:=line "PMMAT2|105005|Adamová, Marie |zk|ESF B-HPS FP [sem 2]"
, ,zadani pro "Adamová, Marie " 105005
pomoci vhodne substituce vypoctete integral
`2*Int((x^2+1)*(x^2+1+2*arctan(x^2+1)*x^4+4*arctan(x^2+1)*x^2+4*arctan(x^2+1))
*x/(x^4+2*x^2+2),x) = (x^2+1)^2*arctan(x^2+1)`
2 ( )+x2
1
( )+ + + +x2
1 2 ( )arctan +x2
1 x4
4 ( )arctan +x2
1 x2
4 ( )arctan +x2
1 x
+ +x4
2 x2
2( ) xd ( )+x2
1
2
( )arctan +x2
1= substituce: =t +x2
1, , ,
=d+
t2
+t2
1
2 t ( )arctan t t ( )arctan t t2
:=line "PMMAT2| 99521|Albrechtová, Kristýna|zk|ESF B-HPS NH [sem 6]"
, ,zadani pro "Albrechtová, Kristýna" 99521
pomoci vhodne substituce vypoctete integral
`Int((cos(ln(x))*ln(x)+sin(ln(x)))/x,x) = sin(ln(x))*ln(x)`
=d
+( )cos ( )ln x ( )ln x ( )sin ( )ln x
x
x ( )sin ( )ln x ( )ln x substituce:, ,
=t ( )ln x =d+( )cos t t ( )sin t t ( )sin t t,
:=line "PMMAT2|100108|Babák, Jan |zk|ESF M-HPS RRS [sem 6]"
, ,zadani pro "Babák, Jan " 100108
pomoci vhodne substituce vypoctete integral
`ln(2)*Int(2^sin(x)*cos(x),x) = 2^sin(x)`
, , ,=( )ln 2 d2
( )sin x
( )cos x x 2
( )sin x
substituce: =t ( )sin x =d2t
( )ln 2 t 2t
line :=
"PMMAT2|174666|Bedná, Martin |zk|ESF M-HPS HOSP [sem 2]"
, ,zadani pro "Bedná, Martin " 174666
pomoci vhodne substituce vypoctete integral
`-2*Int((-x^2-
1+2*arctan(x^2+1)*x^4+4*arctan(x^2+1)*x^2+4*arctan(x^2+1))*x/(x^4+2*x^2+2)/(x^
2+1)^3,x) = arctan(x^2+1)/(x^2+1)^2`
2
( )- - + + +x2
1 2 ( )arctan +x2
1 x4
4 ( )arctan +x2
1 x2
4 ( )arctan +x2
1
( )+ +x4
2 x2
2 ( )+x2
1
3
-
xd
( )arctan +x2
1
( )+x2
1
2
= substituce: =t +x2
1, , ,
=d-
1
( )+t2
1 t2
2 ( )arctan t
t3
t
( )arctan t
t2
line :=
"PMMAT2|174933|Benda, Vladislav |zk|ESF M-EKT EKON [sem 2]"
, ,zadani pro "Benda, Vladislav " 174933
pomoci vhodne substituce vypoctete integral
`2*Int(cos(x^2-1)/sin(x^2-1)*x,x) = ln(sin((x-1)*(x+1)))`
=2 d
( )cos -x2
1 x
( )sin -x2
1
x ( )ln ( )sin ( )-x 1 ( )+x 1 substituce: =t -x2
1, , ,
=d
( )cos t
( )sin t
t ( )ln ( )sin t
:=line "PMMAT2|172164|Beníková, Petra |zk|ESF B-HPS FP [sem 2]"
, ,zadani pro "Beníková, Petra " 172164
pomoci vhodne substituce vypoctete integral
`2*Int(sin(x)*cos(x)/(2-2*cos(x)^2+cos(x)^4),x) = arctan(sin(x)^2)`
=2 d
( )sin x ( )cos x
- +2 2 ( )cos x 2
( )cos x 4
x ( )arctan ( )sin x 2
substituce: =t ( )sin x, , ,
=d
2 t
+t4
1
t ( )arctan t2
:=line "PMMAT2|174769|Blaha, Robert |zk|ESF M-HPS FP [sem 2]"
, ,zadani pro "Blaha, Robert " 174769
pomoci vhodne substituce vypoctete integral
`Int((cos(sin(x))*sin(x)-sin(sin(x)))*cos(x)/sin(x)^2,x) =
sin(sin(x))/sin(x)`
=d
( )-( )cos ( )sin x ( )sin x ( )sin ( )sin x ( )cos x
( )sin x 2
x
( )sin ( )sin x
( )sin x
substituce:, ,
=t ( )sin x =d-
( )cos t
t
( )sin t
t2
t
( )sin t
t
,
:=line "PMMAT2|151092|Cífka, Michal |zk|ESF B-EKM POH [sem 2]"
, ,zadani pro "Cífka, Michal " 151092
pomoci vhodne substituce vypoctete integral
`2*Int((arctan(x^2+1)*x^4+2*arctan(x^2+1)*x^2+2*arctan(x^2+1)+x^2+1)*x/(x^4+2*
x^2+2),x) = (x^2+1)*arctan(x^2+1)`
2
( )+ + + +( )arctan +x2
1 x4
2 ( )arctan +x2
1 x2
2 ( )arctan +x2
1 x2
1 x
+ +x4
2 x2
2
xd ( )+x2
1 ( )arctan +x2
1= substituce: =t +x2
1, , ,
=d+( )arctan t
t
+t2
1
t t ( )arctan t
:=line "PMMAT2|171784|Dianika, Róbert |zk|ESF B-HPS FP [sem 2]"
, ,zadani pro "Dianika, Róbert " 171784
pomoci vhodne substituce vypoctete integral
`2*Int(1/sin(x)*cos(x),x) = 2*ln(sin(x))`
, , ,=2 d
( )cos x
( )sin x
x 2 ( )ln ( )sin x substituce: =t ( )sin x =d
2
t
t 2 ( )ln t
:=line "PMMAT2|136915|Doleel, Tomá |zk|ESF B-HPS NH [sem 4]"
, ,zadani pro "Doleel, Tomá " 136915
pomoci vhodne substituce vypoctete integral
`2*Int(cos(x^2+1)*x,x) = sin(x^2+1)`
=2 d( )cos +x2
1 x x ( )sin +x2
1 substituce: =t +x2
1, , ,
=d( )cos t t ( )sin t
:=line "PMMAT2|171845|Fajtová, Veronika |zk|ESF B-HPS FP [sem 2]"
, ,zadani pro "Fajtová, Veronika " 171845
pomoci vhodne substituce vypoctete integral
`Int((cos(sin(x))*sin(x)-sin(sin(x)))*cos(x)/sin(x)^2,x) =
sin(sin(x))/sin(x)`
=d
( )-( )cos ( )sin x ( )sin x ( )sin ( )sin x ( )cos x
( )sin x 2
x
( )sin ( )sin x
( )sin x
substituce:, ,
=t ( )sin x =d-
( )cos t
t
( )sin t
t2
t
( )sin t
t
,
:=line "PMMAT2|172168|Ferák, Ondrej |zk|ESF B-HPS NH [sem 2]"
, ,zadani pro "Ferák, Ondrej " 172168
pomoci vhodne substituce vypoctete integral
`Int(ln(x)*(ln(x)+2*arctan(ln(x))*ln(x)^2+2*arctan(ln(x)))/(ln(x)^2+1)/x,x) =
arctan(ln(x))*ln(x)^2`
d
( )ln x ( )+ +( )ln x 2 ( )arctan ( )ln x ( )ln x 2
2 ( )arctan ( )ln x
( )+( )ln x 2
1 x
x =
( )arctan ( )ln x ( )ln x 2
substituce: =t ( )ln x, , ,
=d+
t2
+t2
1
2 t ( )arctan t t ( )arctan t t2
:=line "PMMAT2|172186|Florová, Zuzana |zk|ESF B-HPS RRS [sem 2]"
, ,zadani pro "Florová, Zuzana " 172186
pomoci vhodne substituce vypoctete integral
`-Int(1/sin(x)^2*cos(x),x) = 1/sin(x)`
, , ,=- d
( )cos x
( )sin x 2
x
1
( )sin x
substituce: =t ( )sin x =d-
1
t2
t
1
t
:=line "PMMAT2|135083|Havlita, Luká |zk|ESF B-HPS NH [sem 2]"
, ,zadani pro "Havlita, Luká "135083
pomoci vhodne substituce vypoctete integral
`2*Int(ln(x)*exp(ln(x)^2)/x,x) = exp(ln(x)^2)`
, , ,=2 d
( )ln x eeee
( )( )ln x 2
x
x eeee
( )( )ln x 2
substituce: =t ( )ln x =d2 t eeee
( )t2
t eeee
( )t2
:=line "PMMAT2|171776|Holasová, Pavla |zk|ESF B-HPS FP [sem 2]"
, ,zadani pro "Holasová, Pavla " 171776
pomoci vhodne substituce vypoctete integral
`Int(cos(x)*ln(sin(x))+cos(x),x) = sin(x)*ln(sin(x))`
=d+( )cos x ( )ln ( )sin x ( )cos x x ( )sin x ( )ln ( )sin x substituce: =t ( )sin x, , ,[
=d+( )ln t 1 t t ( )ln t ]
:=line "PMMAT2|171762|Hurníková, Tereza |zk|ESF B-HPS FP [sem 2]"
, ,zadani pro "Hurníková, Tereza " 171762
pomoci vhodne substituce vypoctete integral
`Int((cos(ln(x))*ln(x)-sin(ln(x)))/ln(x)^2/x,x) = sin(ln(x))/ln(x)`
=d
-( )cos ( )ln x ( )ln x ( )sin ( )ln x
( )ln x 2
x
x
( )sin ( )ln x
( )ln x
substituce: =t ( )ln x, , ,
=d-
( )cos t
t
( )sin t
t2
t
( )sin t
t
:=line "PMMAT2| 99517|Charvát, Ondej |zk|ESF B-HPS RRS [sem 2]"
, ,zadani pro "Charvát, Ondej " 99517
pomoci vhodne substituce vypoctete integral
`Int(cos(sin(x))*cos(x),x) = sin(sin(x))`
=d( )cos ( )sin x ( )cos x x ( )sin ( )sin x substituce: =t ( )sin x, , ,[
=d( )cos t t ( )sin t ]
line "PMMAT2|174783|Jakubcová, Simona |zk|ESF M-HPS HOSP [\:=
sem 2]"
, ,zadani pro "Jakubcová, Simona " 174783
pomoci vhodne substituce vypoctete integral
`2*Int((x^2+1)*(cos(x^2+1)*x^2+cos(x^2+1)+2*sin(x^2+1))*x,x) =
(x^2+1)^2*sin(x^2+1)`
2 d( )+x2
1 ( )+ +( )cos +x2
1 x2
( )cos +x2
1 2 ( )sin +x2
1 x x =
( )+x2
1
2
( )sin +x2
1 substituce: =t +x2
1, , ,
=d+( )cos t t2
2 ( )sin t t t ( )sin t t2
:=line "PMMAT2| 73899|Jurek, Daniel |zk|ESF B-HPS VEK [sem 6]"
, ,zadani pro "Jurek, Daniel " 73899
pomoci vhodne substituce vypoctete integral
`Int(sin(x)*(cos(sin(x))*sin(x)+2*sin(sin(x)))*cos(x),x) =
sin(sin(x))*sin(x)^2`
d( )sin x ( )+( )cos ( )sin x ( )sin x 2 ( )sin ( )sin x ( )cos x x =
( )sin ( )sin x ( )sin x 2
substituce: =t ( )sin x, , ,
=d+( )cos t t2
2 ( )sin t t t ( )sin t t2
:=line "PMMAT2|171933|Kamenská, Katarína |zk|ESF B-HPS FP [sem 2]"
, ,zadani pro "Kamenská, Katarína " 171933
pomoci vhodne substituce vypoctete integral
`Int((cos(ln(x))*ln(x)+sin(ln(x)))/x,x) = sin(ln(x))*ln(x)`
=d
+( )cos ( )ln x ( )ln x ( )sin ( )ln x
x
x ( )sin ( )ln x ( )ln x substituce:, ,
=t ( )ln x =d+( )cos t t ( )sin t t ( )sin t t,
:=line "PMMAT2|170527|Kantor, Ondej |zk|ESF B-HPS FP [sem 2]"
, ,zadani pro "Kantor, Ondej " 170527
pomoci vhodne substituce vypoctete integral
`2*Int(x/(x^4+2*x^2+2),x) = arctan(x^2+1)`
=2 d
x
+ +x4
2 x2
2
x ( )arctan +x2
1 substituce: =t +x2
1, , , =d
1
+t2
1
t ( )arctan t
line :=
"PMMAT2|174836|Kapoun, Vítzslav |zk|ESF M-HPS VEK [sem 2]"
, ,zadani pro "Kapoun, Vítzslav " 174836
pomoci vhodne substituce vypoctete integral
`4*Int((x^2+1)*x/(x^8+4*x^6+6*x^4+4*x^2+2),x) = arctan(x^4+2*x^2+1)`
=4 d
( )+x2
1 x
+ + + +x8
4 x6
6 x4
4 x2
2
x ( )arctan + +x4
2 x2
1 substituce:, ,
=t +x2
1 =d
2 t
+t4
1
t ( )arctan t2
,
line :=
"PMMAT2|174675|Kedro, Milan |zk|ESF M-HPS HOSP [sem 2]"
, ,zadani pro "Kedro, Milan " 174675
pomoci vhodne substituce vypoctete integral
`2*Int(cos(x^2-1)/sin(x^2-1)*x,x) = ln(sin((x-1)*(x+1)))`
=2 d
( )cos -x2
1 x
( )sin -x2
1
x ( )ln ( )sin ( )-x 1 ( )+x 1 substituce: =t -x2
1, , ,
=d
( )cos t
( )sin t
t ( )ln ( )sin t
:=line "PMMAT2|191617|Klimková, Jana |zk|ESF B-HPS FP [sem 2]"
, ,zadani pro "Klimková, Jana " 191617
pomoci vhodne substituce vypoctete integral
`2*Int(cos(x^2-1)/sin(x^2-1)*x,x) = ln(sin((x-1)*(x+1)))`
=2 d
( )cos -x2
1 x
( )sin -x2
1
x ( )ln ( )sin ( )-x 1 ( )+x 1 substituce: =t -x2
1, , ,
=d
( )cos t
( )sin t
t ( )ln ( )sin t
line :=
"PMMAT2|174818|Kopr, Eduard |zk|ESF M-HPS HOSP [sem 2]"
, ,zadani pro "Kopr, Eduard " 174818
pomoci vhodne substituce vypoctete integral
`4*Int((x^2+1)*exp((x^2+1)^2)*x,x) = exp((x^2+1)^2)`
=4 d( )+x2
1 eeee
( )+x2 1
2
x x eeee
( )+x2 1
2
substituce: =t +x2
1, , , =d2 t eeee
( )t2
t eeee
( )t2
line :=
"PMMAT2|174678|Koíková, Irena |zk|ESF M-EKM POH [sem 2]"
, ,zadani pro "Koíková, Irena " 174678
pomoci vhodne substituce vypoctete integral
`-Int(sin(ln(x))/x,x) = cos(ln(x))`
=- d
( )sin ( )ln x
x
x ( )cos ( )ln x substituce: =t ( )ln x, , ,
=d- ( )sin t t ( )cos t
line :=
"PMMAT2|174797|Kozáková, Barbora |zk|ESF M-HPS RRS [sem 2]"
, ,zadani pro "Kozáková, Barbora " 174797
pomoci vhodne substituce vypoctete integral
`2*Int(1/ln(x)/x,x) = 2*ln(ln(x))`
, , ,=2 d
1
( )ln x x
x 2 ( )ln ( )ln x substituce: =t ( )ln x =d
2
t
t 2 ( )ln t
:=line "PMMAT2| 78782|Kozel, Petr |zk|ESF B-HPS RRS [sem 4]"
, ,zadani pro "Kozel, Petr " 78782
pomoci vhodne substituce vypoctete integral
`-Int(1/(ln(x+1)-1)^2/(x+1),x) = 1/(ln(x+1)-1)`
=- d
1
( )-( )ln +x 1 1 2
( )+x 1
x
1
-( )ln +x 1 1
substituce: =t ( )ln +x 1, , ,
=d-
1
( )-t 1 2
t
1
-t 1
:=line "PMMAT2| 99730|Krková, Marie |zk|ESF B-HPS NH [sem 2]"
, ,zadani pro "Krková, Marie "99730
pomoci vhodne substituce vypoctete integral
`Int((ln(x)-1)/ln(x)^2,x) = x/ln(x)`
, , ,=d
-( )ln x 1
( )ln x 2
x
x
( )ln x
substituce: =t ( )ln x =d-
eeeet
t
eeeet
t2
t
eeeet
t
:=line "PMMAT2|173143|Kuerová, Petra |zk|ESF M-HPS FP [sem 2]"
, ,zadani pro "Kuerová, Petra " 173143
pomoci vhodne substituce vypoctete integral
`Int(sin(x)*(cos(sin(x))*sin(x)+2*sin(sin(x)))*cos(x),x) =
sin(sin(x))*sin(x)^2`
d( )sin x ( )+( )cos ( )sin x ( )sin x 2 ( )sin ( )sin x ( )cos x x =
( )sin ( )sin x ( )sin x 2
substituce: =t ( )sin x, , ,
=d+( )cos t t2
2 ( )sin t t t ( )sin t t2
line :=
"PMMAT2|172059|Kudlová, Monika |zk|ESF B-EKM POH [sem 2]"
, ,zadani pro "Kudlová, Monika " 172059
pomoci vhodne substituce vypoctete integral
`2*Int(exp(x^2+1)*x^3/(x^2+1)^2,x) = exp(x^2+1)/(x^2+1)`
, , ,=2 d
eeee
( )+x
2
1
x3
( )+x2
1
2
x
eeee
( )+x
2
1
+x2
1
substituce: =t +x2
1 =d-
eeeet
t
eeeet
t2
t
eeeet
t
line :=
"PMMAT2|171779|Kusák, Roman |zk|ESF B-EKM POH [sem 2]"
, ,zadani pro "Kusák, Roman " 171779
pomoci vhodne substituce vypoctete integral
`Int((ln(ln(x))+1)/x,x) = ln(x)*ln(ln(x))`
=d
+( )ln ( )ln x 1
x
x ( )ln x ( )ln ( )ln x substituce: =t ( )ln x, , ,
=d+( )ln t 1 t t ( )ln t
:=line "PMMAT2|172078|Lízalová, Eva |zk|ESF B-HPS RRS [sem 2]"
, ,zadani pro "Lízalová, Eva " 172078
pomoci vhodne substituce vypoctete integral
`2*Int(sin(x)*cos(x)/(2-2*cos(x)^2+cos(x)^4),x) = arctan(sin(x)^2)`
=2 d
( )sin x ( )cos x
- +2 2 ( )cos x 2
( )cos x 4
x ( )arctan ( )sin x 2
substituce: =t ( )sin x, , ,
=d
2 t
+t4
1
t ( )arctan t2
:=line "PMMAT2|174665|Lorenc, Jan |zk|ESF M-EKM POH [sem 2]"
, ,zadani pro "Lorenc, Jan " 174665
pomoci vhodne substituce vypoctete integral
`2*Int((x^2+1)*(x^2+1+2*arctan(x^2+1)*x^4+4*arctan(x^2+1)*x^2+4*arctan(x^2+1))
*x/(x^4+2*x^2+2),x) = (x^2+1)^2*arctan(x^2+1)`
2 ( )+x2
1
( )+ + + +x2
1 2 ( )arctan +x2
1 x4
4 ( )arctan +x2
1 x2
4 ( )arctan +x2
1 x
+ +x4
2 x2
2( ) xd ( )+x2
1
2
( )arctan +x2
1= substituce: =t +x2
1, , ,
=d+
t2
+t2
1
2 t ( )arctan t t ( )arctan t t2
:=line "PMMAT2| 99655|Malík, David |zk|ESF M-EKM POH [sem 6]"
, ,zadani pro "Malík, David "99655
pomoci vhodne substituce vypoctete integral
`2*Int(exp(x^2+1)*(x^4+4*x^2+3)*x,x) = exp(x^2+1)*(x^4+2*x^2+1)`
=2 deeee
( )+x2 1
( )+ +x4
4 x2
3 x x eeee
( )+x2 1
( )+ +x4
2 x2
1 substituce:, ,
=t +x2
1 =d+eeeet
t2
2 eeeet
t t eeeet
t2
,
:=line "PMMAT2|137128|Markusík, David |zk|ESF M-HPS FP [sem 4]"
, ,zadani pro "Markusík, David " 137128
pomoci vhodne substituce vypoctete integral
`-Int((-1+2*ln(sin(x)))*cos(x)/sin(x)^3,x) = ln(sin(x))/sin(x)^2`
=- d
( )- +1 2 ( )ln ( )sin x ( )cos x
( )sin x 3
x
( )ln ( )sin x
( )sin x 2
substituce: =t ( )sin x, , ,
=d-
1
t3
2 ( )ln t
t3
t
( )ln t
t2
:=line "PMMAT2|100118|Miklas, David |zk|ESF B-HPS FP [sem 6]"
, ,zadani pro "Miklas, David "100118
pomoci vhodne substituce vypoctete integral
`-Int(1/sin(x)^2*cos(x),x) = 1/sin(x)`
, , ,=- d
( )cos x
( )sin x 2
x
1
( )sin x
substituce: =t ( )sin x =d-
1
t2
t
1
t
line :=
"PMMAT2|137816|Mlynka, Jaroslav |zk|ESF M-HPS HOSP [sem 4]"
, ,zadani pro "Mlynka, Jaroslav " 137816
pomoci vhodne substituce vypoctete integral
`-Int((-1+2*ln(sin(x)))*cos(x)/sin(x)^3,x) = ln(sin(x))/sin(x)^2`
=- d
( )- +1 2 ( )ln ( )sin x ( )cos x
( )sin x 3
x
( )ln ( )sin x
( )sin x 2
substituce: =t ( )sin x, , ,
=d-
1
t3
2 ( )ln t
t3
t
( )ln t
t2
line :=
"PMMAT2|107842|Navrkal, Ondej |zk|ESF M-EKM POH [sem 2]"
, ,zadani pro "Navrkal, Ondej " 107842
pomoci vhodne substituce vypoctete integral
`4*Int((x^2+1)*x/(x^8+4*x^6+6*x^4+4*x^2+2),x) = arctan(x^4+2*x^2+1)`
=4 d
( )+x2
1 x
+ + + +x8
4 x6
6 x4
4 x2
2
x ( )arctan + +x4
2 x2
1 substituce:, ,
=t +x2
1 =d
2 t
+t4
1
t ( )arctan t2
,
line :=
"PMMAT2|174963|Novotný, Michal |zk|ESF M-HPS RRS [sem 2]"
, ,zadani pro "Novotný, Michal " 174963
pomoci vhodne substituce vypoctete integral
`2*Int(1/(x^2+1)*x,x) = ln(x^2+1)`
, , ,=2 d
x
+x2
1
x ( )ln +x2
1 substituce: =t +x2
1 =d
1
t
t ( )ln t
line :=
"PMMAT2|171864|Odehnal, Martin |zk|ESF B-EKM POH [sem 2]"
, ,zadani pro "Odehnal, Martin " 171864
pomoci vhodne substituce vypoctete integral
`Int(1/ln(x)/x,x) = ln(ln(x))`
, , ,=d
1
( )ln x x
x ( )ln ( )ln x substituce: =t ( )ln x =d
1
t
t ( )ln t
line :=
"PMMAT2|174734|Ohnheisrová, Iveta |zk|ESF M-HPS HOSP [sem 2]"
, ,zadani pro "Ohnheisrová, Iveta " 174734
pomoci vhodne substituce vypoctete integral
`2*Int(exp(x^2+1)*(x^2-1)*x/(x^2+1)^3,x) = exp(x^2+1)/(x^2+1)^2`
=2 d
eeee
( )+x
2
1
( )-x2
1 x
( )+x2
1
3
x
eeee
( )+x
2
1
( )+x2
1
2
substituce: =t +x2
1, , , =d-
eeeet
t2
2 eeeet
t3
t
eeeet
t2
:=line "PMMAT2|172037|Petrovi, Martin |zk|ESF B-EKM POH [sem 2]"
, ,zadani pro "Petrovi, Martin " 172037
pomoci vhodne substituce vypoctete integral
`Int((ln(ln(x))+1)/x,x) = ln(x)*ln(ln(x))`
=d
+( )ln ( )ln x 1
x
x ( )ln x ( )ln ( )ln x substituce: =t ( )ln x, , ,
=d+( )ln t 1 t t ( )ln t
:=line "PMMAT2| 99620|Petík, Martin |zk|ESF M-HPS FP [sem 4]"
, ,zadani pro "Petík, Martin " 99620
pomoci vhodne substituce vypoctete integral
`2*Int((cos(x^2+1)*x^2+cos(x^2+1)-2*sin(x^2+1))*x/(x^2+1)^3,x) =
sin(x^2+1)/(x^2+1)^2`
=2 d
( )+ -( )cos +x2
1 x2
( )cos +x2
1 2 ( )sin +x2
1 x
( )+x2
1
3
x
( )sin +x2
1
( )+x2
1
2
,
substituce: =t +x2
1 =d-
( )cos t
t2
2 ( )sin t
t3
t
( )sin t
t2
, ,
line :=
"PMMAT2|171888|Podhradský, Juraj |zk|ESF B-EKM POH [sem 2]"
, ,zadani pro "Podhradský, Juraj " 171888
pomoci vhodne substituce vypoctete integral
`2*Int(sin(x)*exp(sin(x)^2)*cos(x),x) = exp(sin(x)^2)`
=2 d( )sin x eeee
( )( )sin x 2
( )cos x x eeee
( )( )sin x 2
substituce: =t ( )sin x, , ,
=d2 t eeee
( )t2
t eeee
( )t2
line :=
"PMMAT2|170290|Pokorný, Frantiek |zk|ESF M-EKM POH [sem 2]"
, ,zadani pro "Pokorný, Frantiek " 170290
pomoci vhodne substituce vypoctete integral
`Int(cos(sin(x))*cos(x),x) = sin(sin(x))`
=d( )cos ( )sin x ( )cos x x ( )sin ( )sin x substituce: =t ( )sin x, , ,[
=d( )cos t t ( )sin t ]
:=line "PMMAT2|134691|Potoková, Zuzana |zk|ESF M-HPS FP [sem 2]"
, ,zadani pro "Potoková, Zuzana " 134691
pomoci vhodne substituce vypoctete integral
`Int(cos(sin(x))*cos(x),x) = sin(sin(x))`
=d( )cos ( )sin x ( )cos x x ( )sin ( )sin x substituce: =t ( )sin x, , ,[
=d( )cos t t ( )sin t ]
line "PMMAT2|174793|Primová, Andrea |zk|ESF M-EKT EKON [s\:=
em 2]"
, ,zadani pro "Primová, Andrea " 174793
pomoci vhodne substituce vypoctete integral
`-2*Int((-1+ln(x^2+1))*x/(x^2+1)^2,x) = ln(x^2+1)/(x^2+1)`
=-2 d
( )- +1 ( )ln +x2
1 x
( )+x2
1
2
x
( )ln +x2
1
+x2
1
substituce: =t +x2
1, , , =d-
1
t2
( )ln t
t2
t
( )ln t
t
line :=
"PMMAT2|171836|Prodlalová, Linda |zk|ESF B-HPS VEK [sem 2]"
, ,zadani pro "Prodlalová, Linda " 171836
pomoci vhodne substituce vypoctete integral
`Int(1/(arctan(x)^2+1)/(x^2+1),x) = arctan(arctan(x))`
=d
1
( )+( )arctan x 2
1 ( )+x2
1
x ( )arctan ( )arctan x substituce:, ,
=t ( )arctan x =d
1
+t2
1
t ( )arctan t,
:=line "PMMAT2|171818|Rojko, Andrej |zk|ESF B-EKM POH [sem 2]"
, ,zadani pro "Rojko, Andrej " 171818
pomoci vhodne substituce vypoctete integral
`2*Int((arctan(x^2+1)*x^4+2*arctan(x^2+1)*x^2+2*arctan(x^2+1)+x^2+1)*x/(x^4+2*
x^2+2),x) = (x^2+1)*arctan(x^2+1)`
2
( )+ + + +( )arctan +x2
1 x4
2 ( )arctan +x2
1 x2
2 ( )arctan +x2
1 x2
1 x
+ +x4
2 x2
2
xd ( )+x2
1 ( )arctan +x2
1= substituce: =t +x2
1, , ,
=d+( )arctan t
t
+t2
1
t t ( )arctan t
:=line "PMMAT2|171756|Ryek, Matou |zk|ESF B-HPS VEK [sem 2]"
, ,zadani pro "Ryek, Matou " 171756
pomoci vhodne substituce vypoctete integral
`-Int(1/sin(x)^2*cos(x),x) = 1/sin(x)`
, , ,=- d
( )cos x
( )sin x 2
x
1
( )sin x
substituce: =t ( )sin x =d-
1
t2
t
1
t
:=line "PMMAT2|174809|Slezák, Martin |zk|ESF M-EKM POH [sem 2]"
, ,zadani pro "Slezák, Martin "174809
pomoci vhodne substituce vypoctete integral
`2*Int(exp(x^2+1)*x,x) = exp(x^2+1)`
, , ,=2 deeee
( )+x2 1
x x eeee
( )+x2 1
substituce: =t +x2
1 =deeeet
t eeeet
:=line "PMMAT2|171885|Slezáková, Petra |zk|ESF B-HPS VEK [sem 2]"
, ,zadani pro "Slezáková, Petra " 171885
pomoci vhodne substituce vypoctete integral
`2*Int(exp(x^2+1)*x,x) = exp(x^2+1)`
, , ,=2 deeee
( )+x2 1
x x eeee
( )+x2 1
substituce: =t +x2
1 =deeeet
t eeeet
:=line "PMMAT2|171931|Staro, Richard |zk|ESF B-HPS FP [sem 2]"
, ,zadani pro "Staro, Richard " 171931
pomoci vhodne substituce vypoctete integral
`Int((cos(sin(x))*sin(x)-sin(sin(x)))*cos(x)/sin(x)^2,x) =
sin(sin(x))/sin(x)`
=d
( )-( )cos ( )sin x ( )sin x ( )sin ( )sin x ( )cos x
( )sin x 2
x
( )sin ( )sin x
( )sin x
substituce:, ,
=t ( )sin x =d-
( )cos t
t
( )sin t
t2
t
( )sin t
t
,
:=line "PMMAT2|172095|Steiger, Zdenk |zk|ESF B-EKM POH [sem 2]"
, ,zadani pro "Steiger, Zdenk " 172095
pomoci vhodne substituce vypoctete integral
`2*Int(cos(x^2-1)/sin(x^2-1)*x,x) = ln(sin((x-1)*(x+1)))`
=2 d
( )cos -x2
1 x
( )sin -x2
1
x ( )ln ( )sin ( )-x 1 ( )+x 1 substituce: =t -x2
1, , ,
=d
( )cos t
( )sin t
t ( )ln ( )sin t
:=line "PMMAT2|174905|Stratil, Martin |zk|ESF M-EKT EKON [sem 2]"
, ,zadani pro "Stratil, Martin "174905
pomoci vhodne substituce vypoctete integral
`2*Int(exp(x^2+1)*(x^4+4*x^2+3)*x,x) = exp(x^2+1)*(x^4+2*x^2+1)`
=2 deeee
( )+x2 1
( )+ +x4
4 x2
3 x x eeee
( )+x2 1
( )+ +x4
2 x2
1 substituce:, ,
=t +x2
1 =d+eeeet
t2
2 eeeet
t t eeeet
t2
,
:=line "PMMAT2|174905|Stratil, Martin |zk|ESF M-HPS HOSP [sem 2]"
, ,zadani pro "Stratil, Martin "174905
pomoci vhodne substituce vypoctete integral
`2*Int(exp(x^2+1)*(x^4+4*x^2+3)*x,x) = exp(x^2+1)*(x^4+2*x^2+1)`
=2 deeee
( )+x2 1
( )+ +x4
4 x2
3 x x eeee
( )+x2 1
( )+ +x4
2 x2
1 substituce:, ,
=t +x2
1 =d+eeeet
t2
2 eeeet
t t eeeet
t2
,
line :=
"PMMAT2|172083|Svobodová, Veronika |zk|ESF M-HPS FP [sem 2]"
, ,zadani pro "Svobodová, Veronika " 172083
pomoci vhodne substituce vypoctete integral
`-Int((-
ln(x)+2*arctan(ln(x))*ln(x)^2+2*arctan(ln(x)))/(ln(x)^2+1)/ln(x)^3/x,x) =
arctan(ln(x))/ln(x)^2`
=- d
- + +( )ln x 2 ( )arctan ( )ln x ( )ln x 2
2 ( )arctan ( )ln x
( )+( )ln x 2
1 ( )ln x 3
x
x
( )arctan ( )ln x
( )ln x 2
,
substituce: =t ( )ln x =d-
1
( )+t2
1 t2
2 ( )arctan t
t3
t
( )arctan t
t2
, ,
:=line "PMMAT2|174671|afáová, Monika |zk|ESF M-HPS FP [sem 2]"
, ,zadani pro "afáová, Monika " 174671
pomoci vhodne substituce vypoctete integral
`ln(2)*Int(2^(x^2+1)*x,x) = 2^(x^2)`
, , ,=( )ln 2 d2
( )+x2 1
x x 2
( )x2
substituce: =t x2
=d2t
( )ln 2 t 2t
line :=
"PMMAT2| 99492|amlová, Markéta |zk|ESF M-HPS RRS [sem 6]"
, ,zadani pro "amlová, Markéta " 99492
pomoci vhodne substituce vypoctete integral
`2*Int((arctan(x^2+1)*x^4+2*arctan(x^2+1)*x^2+2*arctan(x^2+1)+x^2+1)*x/(x^4+2*
x^2+2),x) = (x^2+1)*arctan(x^2+1)`
2
( )+ + + +( )arctan +x2
1 x4
2 ( )arctan +x2
1 x2
2 ( )arctan +x2
1 x2
1 x
+ +x4
2 x2
2
xd ( )+x2
1 ( )arctan +x2
1= substituce: =t +x2
1, , ,
=d+( )arctan t
t
+t2
1
t t ( )arctan t
line :=
"PMMAT2|172194|auerová, Ludmila |zk|ESF B-EKM POH [sem 2]"
, ,zadani pro "auerová, Ludmila " 172194
pomoci vhodne substituce vypoctete integral
`2*Int((cos(x^2+1)*x^2+cos(x^2+1)+sin(x^2+1))*x,x) = (x^2+1)*sin(x^2+1)`
=2 d( )+ +( )cos +x2
1 x2
( )cos +x2
1 ( )sin +x2
1 x x ( )+x2
1 ( )sin +x2
1 ,
substituce: =t +x2
1 =d+( )cos t t ( )sin t t ( )sin t t, ,
:=line "PMMAT2|172149|erý, Martin |zk|ESF B-HPS FP [sem 2]"
, ,zadani pro "erý, Martin " 172149
pomoci vhodne substituce vypoctete integral
`2*Int(ln(x)*exp(ln(x)^2)/x,x) = exp(ln(x)^2)`
, , ,=2 d
( )ln x eeee
( )( )ln x 2
x
x eeee
( )( )ln x 2
substituce: =t ( )ln x =d2 t eeee
( )t2
t eeee
( )t2
line :=
"PMMAT2|170179|mídová, Lucie |zk|ESF M-EKM POH [sem 2]"
, ,zadani pro "mídová, Lucie " 170179
pomoci vhodne substituce vypoctete integral
`-Int((sin(x)-2*arctan(sin(x))+arctan(sin(x))*cos(x)^2)*cos(x)/sin(x)^2/(-
2+cos(x)^2),x) = 1/sin(x)*arctan(sin(x))`
- d
( )- +( )sin x 2 ( )arctan ( )sin x ( )arctan ( )sin x ( )cos x 2
( )cos x
( )sin x 2
( )- +2 ( )cos x 2
x =
( )arctan ( )sin x
( )sin x
substituce: =t ( )sin x, , ,
=d-
1
( )+t2
1 t
( )arctan t
t2
t
( )arctan t
t
:=line "PMMAT2|171979|astná, Pavlína |zk|ESF B-HPS VEK [sem 2]"
, ,zadani pro "astná, Pavlína " 171979
pomoci vhodne substituce vypoctete integral
`Int((ln(x)-2)/ln(x)^3,x) = x/ln(x)^2`
, , ,=d
-( )ln x 2
( )ln x 3
x
x
( )ln x 2
substituce: =t ( )ln x =d-
eeeet
t2
2 eeeet
t3
t
eeeet
t2
line :=
"PMMAT2|106163|várová, Jana |zk|ESF M-EKT EKON [sem 2]"
, ,zadani pro "várová, Jana " 106163
pomoci vhodne substituce vypoctete integral
`2*Int(1/(x^2+1)*x,x) = ln(x^2+1)`
, , ,=2 d
x
+x2
1
x ( )ln +x2
1 substituce: =t +x2
1 =d
1
t
t ( )ln t
line :=
"PMMAT2|172008|Tomková, Hana |zk|ESF B-HPS VEK [sem 2]"
, ,zadani pro "Tomková, Hana " 172008
pomoci vhodne substituce vypoctete integral
`Int(1/(arctan(x)^2+1)/(x^2+1),x) = arctan(arctan(x))`
=d
1
( )+( )arctan x 2
1 ( )+x2
1
x ( )arctan ( )arctan x substituce:, ,
=t ( )arctan x =d
1
+t2
1
t ( )arctan t,
line :=
"PMMAT2|171930|Turcsányi, Richard |zk|ESF B-EKM POH [sem 2]"
, ,zadani pro "Turcsányi, Richard " 171930
pomoci vhodne substituce vypoctete integral
`Int((cos(sin(x))*sin(x)+sin(sin(x)))*cos(x),x) = sin(x)*sin(sin(x))`
=d( )+( )cos ( )sin x ( )sin x ( )sin ( )sin x ( )cos x x ( )sin x ( )sin ( )sin x ,[
substituce: =t ( )sin x =d+( )cos t t ( )sin t t ( )sin t t, , ]
:=line "PMMAT2|171975|Turková, Lenka |zk|ESF B-HPS RRS [sem 2]"
, ,zadani pro "Turková, Lenka " 171975
pomoci vhodne substituce vypoctete integral
`Int(exp(sin(x))*sin(x)*(sin(x)+2)*cos(x),x) = exp(sin(x))*sin(x)^2`
=deeee
( )sin x
( )sin x ( )+( )sin x 2 ( )cos x x eeee
( )sin x
( )sin x 2
substituce:, ,
=t ( )sin x =d+eeeet
t2
2 eeeet
t t eeeet
t2
,
:=line "PMMAT2| 65353|Valentová, Jitka |zk|ESF M-HPS VEK [sem 4]"
, ,zadani pro "Valentová, Jitka " 65353
pomoci vhodne substituce vypoctete integral
`Int(sin(x)*(-sin(x)-4*arctan(sin(x))+2*arctan(sin(x))*cos(x)^2)*cos(x)/(-
2+cos(x)^2),x) = arctan(sin(x))*sin(x)^2`
( )sin x ( )- - +( )sin x 4 ( )arctan ( )sin x 2 ( )arctan ( )sin x ( )cos x 2
( )cos x
- +2 ( )cos x 2
d
x ( )arctan ( )sin x ( )sin x 2
= substituce: =t ( )sin x, , ,
=d+
t2
+t2
1
2 t ( )arctan t t ( )arctan t t2
line :=
"PMMAT2|171857|Valentová, Lenka |zk|ESF B-EKM POH [sem 2]"
, ,zadani pro "Valentová, Lenka " 171857
pomoci vhodne substituce vypoctete integral
`Int((cos(ln(x))*ln(x)-2*sin(ln(x)))/ln(x)^3/x,x) = sin(ln(x))/ln(x)^2`
=d
-( )cos ( )ln x ( )ln x 2 ( )sin ( )ln x
( )ln x 3
x
x
( )sin ( )ln x
( )ln x 2
substituce: =t ( )ln x, , ,
=d-
( )cos t
t2
2 ( )sin t
t3
t
( )sin t
t2
:=line "PMMAT2|174790|Váa, Vladislav |zk|ESF M-HPS FP [sem 2]"
, ,zadani pro "Váa, Vladislav " 174790
pomoci vhodne substituce vypoctete integral
`-Int((-1+ln(ln(x)))/ln(x)^2/x,x) = ln(ln(x))/ln(x)`
=- d
- +1 ( )ln ( )ln x
( )ln x 2
x
x
( )ln ( )ln x
( )ln x
substituce: =t ( )ln x, , , =d-
1
t2
( )ln t
t2
t
( )ln t
t
line :=
"PMMAT2|174973|Vdovec, Milan |zk|ESF M-EKM POH [sem 2]"
, ,zadani pro "Vdovec, Milan " 174973
pomoci vhodne substituce vypoctete integral
`2*Int((x^2+1+2*ln(x^2+1)*x^2+2*ln(x^2+1))*x,x) =
x^4*ln(x^2+1)+ln(x^2+1)+2*ln(x^2+1)*x^2-2`
2 d( )+ + +x2
1 2 ( )ln +x2
1 x2
2 ( )ln +x2
1 x x =
+ + -x4
( )ln +x2
1 ( )ln +x2
1 2 ( )ln +x2
1 x2
2 substituce: =t +x2
1, , ,
=d+t 2 t ( )ln t t t2
( )ln t
:=line "PMMAT2|106541|Vegrichtová, Marta |zk|ESF M-HPS FP [sem 2]"
, ,zadani pro "Vegrichtová, Marta " 106541
pomoci vhodne substituce vypoctete integral
`Int((arctan(ln(x))*ln(x)^2+arctan(ln(x))+ln(x))/(ln(x)^2+1)/x,x) =
ln(x)*arctan(ln(x))`
=d
+ +( )arctan ( )ln x ( )ln x 2
( )arctan ( )ln x ( )ln x
( )+( )ln x 2
1 x
x ( )ln x ( )arctan ( )ln x ,
substituce: =t ( )ln x =d+( )arctan t
t
+t2
1
t t ( )arctan t, ,
:=line "PMMAT2|171976|Virglová, Lucie |zk|ESF B-EKM POH [sem 2]"
, ,zadani pro "Virglová, Lucie "171976
pomoci vhodne substituce vypoctete integral
`Int(exp(sin(x))*(sin(x)-2)*cos(x)/sin(x)^3,x) = exp(sin(x))/sin(x)^2`
=d
eeee
( )sin x
( )-( )sin x 2 ( )cos x
( )sin x 3
x
eeee
( )sin x
( )sin x 2
substituce: =t ( )sin x, , , =d-
eeeet
t2
2 eeeet
t3
t
eeeet
t2
line "PMMAT2|174214|Vojtková, Ludmila |zk|ESF M-EKM POH [s\:=
em 2]"
, ,zadani pro "Vojtková, Ludmila " 174214
pomoci vhodne substituce vypoctete integral
`Int(ln(x)*(ln(x)+2),x) = ln(x)^2*x`
=d( )ln x ( )+( )ln x 2 x ( )ln x 2
x substituce: =t ( )ln x, , ,
=d+eeeet
t2
2 eeeet
t t eeeet
t2
:=line "PMMAT2|172170|Vravko, Matej |zk|ESF B-HPS RRS [sem 2]"
, ,zadani pro "Vravko, Matej " 172170
pomoci vhodne substituce vypoctete integral
`4*Int((x^2+1)*x/(x^8+4*x^6+6*x^4+4*x^2+2),x) = arctan(x^4+2*x^2+1)`
=4 d
( )+x2
1 x
+ + + +x8
4 x6
6 x4
4 x2
2
x ( )arctan + +x4
2 x2
1 substituce:, ,
=t +x2
1 =d
2 t
+t4
1
t ( )arctan t2
,
:=line "PMMAT2|171839|Zlato, Michal |zk|ESF B-EKM POH [sem 2]"
, ,zadani pro "Zlato, Michal "171839
pomoci vhodne substituce vypoctete integral
`Int(1/ln(x)/x,x) = ln(ln(x))`
, , ,=d
1
( )ln x x
x ( )ln ( )ln x substituce: =t ( )ln x =d
1
t
t ( )ln t
:=line "PMMAT2|174990|Zubatý, Adam |zk|ESF M-HPS FP [sem 2]"
, ,zadani pro "Zubatý, Adam " 174990
pomoci vhodne substituce vypoctete integral
`4*Int((x^2+1)*exp((x^2+1)^2)*x,x) = exp((x^2+1)^2)`
=4 d( )+x2
1 eeee
( )+x2 1
2
x x eeee
( )+x2 1
2
substituce: =t +x2
1, , , =d2 t eeee
( )t2
t eeee
( )t2
:=line 0