MATH141 TUT 3
The tutorial questions for each week are recommended problems. It is in your own interest
to work out as many as of the remaining problems of the exercises in the prescribed textbook
out as possible.
Standard Integrals and u-Substitution - Exercise 7.1
1. Evaluate the following integrals by making appropriate u-substitution and applying
standard integrals:
(a)
Z
cot2 x + 1dx
(b)
Z
2x + 1
dx
x2 + 1
(c)
Z
e2x+1 dx
(d)
Z
1
dx
(ex + e−x )2
(e)
Z
1
dx
csc x
Z
1
dx
cos2 x
(f)
(g)
Z
(6 − 2x)2 dx
(h)
Z
√
4 + 9xdx
(i)
Z
2x sec(x2 )dx
(j)
Z
√
cos x sin xdx
1
(k)
Z
cos 3x
dx
2 + sin 3x
(l)
Z
x tan2 (x2 ) sec2 (x2 )dx
(m)
Z
e3 tan x sec2 xdx
(n)
Z
x
√
dx
4 1 − x2
(o)
Z
cos x
√ 2
dx
sin x sin x + 4
(p)
Z
√
ex
dx
9 + e2x
(q)
−1
Z
etan x
dx
1 + x2
Z
e x−2
√
dx
x−2
(r)
√
(s)
Z
dx
dx
x ln x
(t)
Z
dx
√ √x dx
x5
Z
sin(ln x)
dx
x
Z
ex
√
dx
9 − e2x
(u)
(v)
(w)
Z
2
2
x5−x dx
2. (a) Derive the identity
sec2 x
1
=
.
tan x
sin x cos x
(b) Using the identity sin 2x = 2 sin x cos x with (a), evaluate
Z
csc xdx.
Integration by Parts - Exercise 7.2
3. Evaluate the following integrals:
(a)
Z
xe−x dx
(b)
Z
2x2 e−2x dx
(c)
Z
x sin 5xdx
(d)
Z
2x2 cos xdx
(e)
Z
2x ln xdx
(f)
√
5 x ln xdx
Z
(g)
Z
(ln x)2 dx
(h)
Z
e−x sin xdx
(i)
Z
e3x cos 2xdx
(j)
Z
sin(ln x)dx
3
(k)
Z
x
sec2 xdx
3
Z
xex
dx
(x + 1)2
(l)
(m)
e
Z
x3 ln xdx
1
(n)
e
Z
√
e
ln x
dx
x2
(o)
4
Z
sec−1
√
xdx
2
(p)
2
Z
x sec−1 xdx
1
(q)
3
Z
√
√
x sec−1 xdx
1
4. Evaluate the following integrals by making a u-substitution and then integrating by
parts:
(a)
Z
√
e
x
dx
(b)
Z
√
sin xdx
5. Evaluate the integral
Z
1
√
0
(a) using integration by parts
(b) using the substitution u =
√
x2 + 1.
6. Derive the following reduction formula:
4
x3
dx
x2 + 1
(a)
Z
secn−2 x tan x n − 2
+
sec xdx =
n−1
n−1
n
Z
secn−2 xdx
(b)
Z
tann−1 x
−
tan xdx =
n−1
n
Z
tann−2 xdx
(c)
Z
n x
Z
n x
x e dx = x e − n!
xn−1 ex dx
7. Use reduction formula to evaluate the following integrals:
(a)
Z
sin4 xdx
(b)
Z
sin5 xdx
(c)
Z
cos5 xdx
(d)
Z
cos6 xdx
(e)
Z
tan4 xdx
(f)
Z
sec4 xdx
(g)
Z
x3 ex dx
(h)
Z
x2 e2x dx
(i)
Z
5
xe−
√
x
dx