Practice Integrals
If you would like answers, use Wolfram alpha (just type in, e.g., “Integrate x2 + 1” and it should give
3
you x3 + x + C). If you get stuck on how an integral works, there are some hints on the back.
1.
Z
2.
Z
3.
Z
xex dx
Z
Z
ln(t)dt
4.
Z
(p + 1)3 (p + 2)2 dp
Z
1
x ln(x)
Z
Z
Z
7xe
−2x2
Z
2
Z
Z
x sin
2
Z
5y
dx
y 2
4
1
dx
x3
θ sin(π + θ2 )dθ
x ln(x)dx
x
Z
t2 e−t dt
Z
cos(s )sds
dx
Z
cos(x) sin(2x)dx
ecos(5y) sin(5y)dy
2
(1 + s) cos(s)ds
Z
ex cos(x)dx
(sin(y) + 1)2 dy
x
(x2 − 1)3/2
Z
5.
Z
(r + 2)5/2 dr
−1
2
x3 ex dx
dy
6.
Z
e
−x
Z cos(5x)dx
1
x + 5x − x −
2
3
2
x6 − x4 − x2
dx
x2
Z
dx
Z
7.
Z
Z
3 5x
(5x) e dx
8.
Z
(u + 1)du
1
Z
xm dx
0
Z
0
11.
1
p
1 − x2 dx
Z
(1 + t2 )−3/2 dt
9.
10.
Z
2
Z
Z
1
du
u2 + 1
Z
x+1
dx
x
∞
e−x dx
x−µ 2
1
√ e−( σ ) dx
σ 2π
−∞
Z
x
Z
ln(t)dt
1
1
Z
√
0
x
2
0
1
Z
tan(x)dx
π/4
−π/4
Z
5
xdx
3e−
x2
2
Z
xdx
0
Z
et dt
(x2 + 1)3/2
tan(x)dx
−∞
0
∞
Z
2
re−r dr
x
ln(x)(x − 1)
dx
x
1
2x
Z
1
√
dx
1 − x2
sin(θ)
dθ
cos2 (θ) + 1
∞
Z
Z
Z
ln(y)
dy
y
1
ds
1 + s2
0
x
e−y dy.
Z
cos(t)dt
0
Z
x
1
dx
0
Selected Hints/Answers
1.
R
xex is integration by parts. ln(t)dt is also integration by parts:
Z Z
Z
d
d
t dt = t ln(t) −
ln(t) tdt.
ln(t) = ln(t)
dt
dt
R
ex cos(x)dx is integration by parts, twice:
Z
Z Z
Z
d x
d
x
x
x
x
e cos(x)dx =
e cos(x)dx = e cos(x) − e
cos(x) dx = e cos(x) + ex sin(x)
dx
dx
Z d x
x
= e cos(x) +
e sin(x)dx
dx
Z
Z
d
sin(x)
= ex cos(x) + ex sin(x) − ex cos(x)
= ex cos(x) + ex sin(x) − ex
dx
so
Z
Z
ex cos(x)dx = ex cos(x) + ex sin(x) − ex cos(x)dx
R
so
Z
2
so
For
Z
ex cos(x)dx = ex cos(x) + ex sin(x)
ex cos(x)dx =
1 x
(e cos(x) + ex sin(x)) + C.
2
R
cos(x) sin(2x)dx, note that sin(2x) = 2 sin(x) cos(x) (double angle formula for sine).
R
2. To integrate sin2 (y), use the half angle formula for sine. t2 e−t dt is integration by parts.
3. The first is by parts, the next are all u-substitutions.
2
2
4. For the last one, do it by parts but split it like x3 ex = x2 (xex ).
5. Second one is integration by parts. The others can be done with u-substitutions.
8. I think
something to the numerator.
R the first one is hard. I did it by parts after adding and subtracting
sin(θ)
dθ,
use
u
=
cos(θ).
For
the
last
one,
note
that
tan(θ)
=
For cossin(θ)
2 (θ)+1
cos(θ) and do a u-substitution.
2