with(student): PROBLEMA 281 > Int((3*x+1)^2,x)=int((3*x+1)^2,x

> with(student):
PROBLEMA 281
> Int((3*x+1)^2,x)=int((3*x+1)^2,x);
1
⌠
( 3 x + 1 )2 dx = ( 3 x + 1 )3
⌡
9
> Int(3*x/sqrt(x^2+1),x)=int(3*x/sqrt(x^2+1),x);
⌠

x
3
dx = 3 x 2 + 1

2

x +1
⌡
> Int(sin(x)*cos(x),x)=int(sin(x)*cos(x),x);
1
⌠
sin( x ) cos( x ) dx = − cos( x )2
⌡
2
> Int(ln(x)/x,x)=int(ln(x)/x,x);
⌠
1
 ln( x )

dx = ln( x )2

2
 x
⌡
> Int(1/(4+x^2),x)=int(1/(4+x^2),x);
⌠
 1
1
1 

 x 
d
x
=
arctan

2
2 
 4 + x2
⌡
> Int(1/x*ln(x),x)=int(1/x*ln(x),x);
⌠
1
 ln( x )

d
=
x
ln( x )2

2
 x
⌡
> Int((x+1)/(x^2-1),x)=int((x+1)/(x^2-1),x);Para nosotros ln(|x-1|).
⌠
x+1

dx = ln( x − 1 )

 x2 − 1
⌡
> simplify(Int(1/((x-1)^2+2),x)=int(1/((x-1)^2+2),x));
⌠

1
1
1


 ( x − 1 ) 2 
d
=
x
2
arctan

2
2

 x2 − 2 x + 3
⌡
> Int(exp(sqrt(x))/sqrt(x),x)=int(exp(sqrt(x))/sqrt(x),x);
⌠ ( x)
e
( x)

dx = 2 e


 x
⌡
> Int(arctan(x)/(1+x^2),x)=int(arctan(x)/(1+x^2),x);
⌠
 arctan( x )
1

dx = arctan( x )2
2

2
 x +1
⌡
> Int(x*sqrt(x^2+1),x)=int(x*sqrt(x^2+1),x);
(3 / 2)
⌠
1 2
2

x x + 1 dx = ( x + 1 )
3
⌡
> Int((x^3-1)/(x-1),x)=int((x^3-1)/(x-1),x);
⌠ 3
x − 1
1 3 1 2

d
x
=
x + x +x

3
2
x−1
⌡
> Int(sin(x)/cos(x)^2,x)=int(sin(x)/cos(x)^2,x);
⌠
 sin( x )
1

dx =
2

cos( x )
 cos( x )
⌡
> Int(exp(2*ln(x)),x)=int(exp(2*ln(x)),x);
1
⌠ 2
x dx = x3
⌡
3
> Int(exp(x)/sqrt(1-exp(x)),x)=int(exp(x)/sqrt(1-exp(x)),x);
⌠
 ex

dx = −2

 1 − ex
⌡
1 − ex
PROBLEMA 282
> restart:with(student):
> Int(ln(x),x)=value(intparts(Int(ln(x),x),ln(x)));
⌠
ln( x ) dx = ln( x ) x − x
⌡
> Int(x^2*exp(x),x)=value(intparts(Int(x^2*exp(x),x),x));
⌠ 2 x
x e dx = x ( x ex − ex ) − x ex + 2 ex
⌡
> Int(x^2*ln(x),x)=value(intparts(Int(x^2*ln(x),x),x^2));
2
8
⌠ 2
x ln( x ) dx = x2 ( ln( x ) x − x ) − x3 ln( x ) + x3
⌡
3
9
> Int((x*arcsin(x))/sqrt(1-x^2),x)=value(intparts(Int((x*arcsin(x)
)/sqrt(1-x^2),x),arcsin(x)));
⌠
 x arcsin( x )

dx = −arcsin( x )

2

1
−
x
⌡
1 − x2 + x
> Int(arctan(x),x)=value(intparts(Int(arctan(x),x),arctan(x)));
1
⌠
arctan( x ) dx = arctan( x ) x − ln( 1 + x2 )
⌡
2
> Int(arcsin(x),x)=value(intparts(Int(arcsin(x),x),arcsin(x)));
⌠
arcsin( x ) dx = arcsin( x ) x + 1 − x2
⌡
> Int(ln(x)^2,x)=value(intparts(Int(ln(x)^2,x),ln(x)^2));
⌠
ln( x )2 dx = ln( x )2 x − 2 ln( x ) x + 2 x
⌡
> Int(cos(ln(x)),x)=value(intparts(Int(cos(ln(x)),x),cos(ln(x))));
1
1
⌠
cos( ln( x ) ) dx = cos( ln( x ) ) x + sin( ln( x ) ) x
⌡
2
2
> Int(x*sin(x),x)=value(intparts(Int(x*sin(x),x),x));
⌠
x sin( x ) dx = −x cos( x ) + sin( x )
⌡
> Int(exp(x)*cos(x),x)=value(intparts(Int(exp(x)*cos(x),x),cos(x))
);
1
1
⌠ x
e cos( x ) dx = ex cos( x ) + sin( x ) ex
⌡
2
2
> Int(x^2*cos(x),x)=value(intparts(Int(x^2*cos(x),x),x^2));
⌠ 2
x cos( x ) dx = x2 sin( x ) − 2 sin( x ) + 2 x cos( x )
⌡
> Int((x^2-1)*exp(2*x),x)=value(intparts(Int((x^2-1)*exp(2*x),x),(
x^2-1)));
⌠
1
1 (2 x) 1 (2 x)
(2 x)
(2 x)
2

− xe
+ e
dx = ( − 1 + x 2 ) e
( −1 + x ) e
2
2
4
⌡
PROBLEMA 284
> Int(sin(x)^2,x)=int(sin(x)^2,x);
1
1
⌠
sin( x )2 dx = − cos( x ) sin( x ) + x
⌡
2
2
> %=x/2-sin(2*x)/4;
1
1  1
1
⌠
 sin( x )2 dx = − cos( x ) sin( x ) + x  = x − sin( 2 x )
2
2  2
4
⌡
> Int(sin(t)^2*cos(t)^2,t)=int(sin(t)^2*cos(t)^2,t);
1
1
1
⌠
sin( t )2 cos( t )2 dt = − sin( t ) cos( t )3 + cos( t ) sin( t ) + t
⌡
4
8
8
> Int(sin(x)^3,x)=int(sin(x)^3,x);
1
2
⌠
sin( x )3 dx = − sin( x )2 cos( x ) − cos( x )
⌡
3
3
> Int(sin(t)^3*cos(t)^2,t)=int(sin(t)^3*cos(t)^2,t);
1
2
⌠
sin( t )3 cos( t )2 dt = − sin( t )2 cos( t )3 −
cos( t )3
⌡
5
15
> Int(cos(3*x)*cos(x)^3,x)=int(cos(3*x)*cos(x)^3,x);
1
1
3
3
⌠
cos( 3 x ) cos( x )3 dx =
sin( 6 x ) + x +
sin( 2 x ) +
sin( 4 x )
⌡
48
8
16
32
> Int(sin(x)^5*cos(x),x)=int(sin(x)^5*cos(x),x);
1
⌠
sin( x )5 cos( x ) dx = sin( x )6
⌡
6
> Int(tan(x),x)=int(tan(x),x);
⌠
tan( x ) dx = −ln( cos( x ) )
⌡
> Int(cos(x)^4,x)=int(cos(x)^4,x);
1
3
3
⌠
cos( x )4 dx = sin( x ) cos( x )3 + cos( x ) sin( x ) + x
⌡
4
8
8
> Int(sin(t)*cos(t)^3,t)=int(sin(t)*cos(t)^3,t);
1
⌠
sin( t ) cos( t )3 dt = − cos( t )4
⌡
4
> Int(sin(x/2)*cos(x/3),x)=int(sin(x/2)*cos(x/3),x);
⌠
1 
3
5 
1 
 1 
sin x  cos x  dx = − cos x  − 3 cos x 

3 
5
6 
6 
 2 
⌡
PROBLEMA 286
> Int(exp(x)*cos(exp(x)),x)=int(exp(x)*cos(exp(x)),x);
⌠ x
e cos( ex ) dx = sin( ex )
⌡
> Int(sin(2*x)*cos(2*x)^3,x)=int(sin(2*x)*cos(2*x)^3,x);
1
⌠
sin( 2 x ) cos( 2 x )3 dx = − cos( 2 x )4
⌡
8
> Int(1/sqrt(7+8*x^2),x)=int(1/sqrt(7+8*x^2),x);
⌠

1
1
2


dx =
2 arcsinh
14 x 

4
7

 7 + 8 x2
⌡
> Int(sin(ln(x))/x,x)=int(sin(ln(x))/x,x);
⌠
 sin( ln( x ) )

dx = −cos( ln( x ) )

x

⌡
> Int(x*exp(-x^2),x)=int(x*exp(-x^2),x);
⌠ ( −x2 )
1 ( −x2 )
x e
dx = − e

2
⌡
> Int(ln(x)/sqrt(x),x)=int(ln(x)/sqrt(x),x);
⌠
 ln( x )

dx = 2 x ln( x ) − 4 x

 x
⌡
> Int(ln(x)^3/x,x)=int(ln(x)^3/x,x);
⌠
 ln( x )3
1

dx = ln( x )4

4
 x
⌡
> Int(tan(x)^2,x)=int(tan(x)^2,x);
⌠
tan( x )2 dx = tan( x ) − arctan( tan( x ) )
⌡
> Int((sqrt(x)+ln(x))/x,x)=int((sqrt(x)+ln(x))/x,x);
⌠
 x + ln( x )
1

d
=
x
x
2
+
ln( x )2

x
2

⌡
> Int(x^2/(x^2+2),x)=int(x^2/(x^2+2),x);
⌠ 2
 x
1




x
x
−
d
=
2
arctan
x
2

 2

2

x
+
2

⌡
> Int(exp(2*x)/sqrt(exp(x)+1),x)=int(exp(2*x)/sqrt(exp(x)+1),x);
⌠ (2 x)
 e
(3 / 2)
2

dx = ( e x + 1 )
− 2 ex + 1
 x
3

 e +1
⌡
> Int(exp(sqrt(x)),x)=int(exp(sqrt(x)),x);
⌠ ( x)
( x)
( x)
e
dx = 2 e
x −2e

⌡
> Int(arcsin(x)/x^2,x)=int(arcsin(x)/x^2,x);
⌠
 arcsin( x )
arcsin( x )
1




dx = −
− arctanh
2

2 
x

x
1
x

−

⌡
> Int(1/(x*sqrt(1-ln(x)^2)),x)=int(1/(x*sqrt(1-ln(x)^2)),x);
⌠

1

dx = arcsin( ln( x ) )

 x 1 − ln( x )2
⌡
> Int(3^x*exp(x),x)=int(3^x*exp(x),x);
x
( x ln( 3 ) )
e e
⌠ x x
3 e dx =
⌡
ln( 3 ) + 1
> Int(sqrt(arcsin(x))/sqrt((1-x^2)),x)=int(sqrt(arcsin(x))/sqrt((1
-x^2)),x);
⌠
 arcsin( x )
2
(3 / 2)

dx = arcsin( x )

3

1 − x2
⌡
> Int(x/sqrt(x^2+1),x)=int(x/sqrt(x^2+1),x);
⌠

x

dx =

 1 + x2
⌡
> Int(x/exp(x),x)=int(x/exp(x),x);
1 + x2
⌠
x
x
1
 dx = − −

 ex
ex ex
⌡
> Int(ln(ln(x))/x,x)=int(ln(ln(x))/x,x);
⌠
 ln( ln( x ) )

dx = ln( x ) ln( ln( x ) ) − ln( x )

x

⌡
> Int(x^2/(1+25*x^6),x)=int(x^2/(1+25*x^6),x);
⌠

x2
1

arctan( 5 x3 )
dx =

6
15
 1 + 25 x
⌡