iGCSE (2021 Edition)

# 12.05 Integration of trigonometric functions

Worksheet
Antiderivatives of trigonometric functions
1

Find the primitive function of the following:

a

f'(x)=7 \sin x

b

f'(x)=- 7 \cos x

c

f'(x)=6 \sin x - \cos x

d

f'(x)=5 \sin \left(\dfrac{x}{3}\right) - 2 \cos \left(\dfrac{x}{4}\right)

2

Find the following indefinite integrals:

a

\int \sin 7 x \ dx

b

\int \cos 6 x \ dx

c

\int \sin \left(\dfrac{x}{3}\right)dx

d

\int 9 \sin 3 x \ dx

e

\int - 5 \cos \left(\dfrac{x}{4}\right)dx

f

\int \cos \left(x + \dfrac{\pi}{6}\right)dx

g

\int \sin \left(x - \dfrac{\pi}{6}\right)dx

h

\int \sin \left( 5 x - \dfrac{\pi}{4}\right)dx

i

\int \cos \left(\dfrac{x}{8} + \dfrac{\pi}{3}\right)dx

j

\int 8 \cos \left( 4 x - \dfrac{\pi}{3}\right) dx

k

\int - 8 \sin \left( 2 x + \dfrac{\pi}{6}\right)dx

l

\int 2 \sin \left(\dfrac{x}{7} - \dfrac{\pi}{4}\right)dx

m

\int - 4 \cos \left(\dfrac{x}{7} + \dfrac{\pi}{4}\right)dx

n

\int \left(x^{\frac{1}{4}} + \sin 3 x\right) dx

o

\int \left(\cos x - \sin 3 x\right) dx

p

\int \left(\sin \left(\dfrac{x}{3}\right) + \cos \left(\dfrac{x}{3}\right)\right) dx

q

\int 7 \left(\sin 7 x + 9 \cos 3 x\right)dx

r

\int \cos x \left(4 + \tan x\right) dx

3

Consider the function y=x \sin x.

a

Find \dfrac{d}{dx} \left( x \sin x\right).

b

Hence find \int 4 x \cos x \,dx.

Equations of functions
4

Given the gradient function and a point on the curve, find y in terms of x:

a

\dfrac{d y}{d x} = - 3 \sin \left(\dfrac{x}{2}\right) and \left(2 \pi, 5 \right).

b

\dfrac{d y}{d x} = \sin 2 x + \cos 3 x and \left(\dfrac{\pi}{2}, 4 \right).

5

Consider f \rq \left( x \right) = k \cos 3 x for some constant k, with f \rq \left( 0 \right) = 2 and f \left( \dfrac{\pi}{6} \right) = 6.

a

Find the value of k.

b

Hence find f \left( x \right).

6

Consider f \rq \left( x \right) = k \sin \left(\dfrac{x}{4}\right) for some constant k, with f \left( 2 \pi \right) = 1 and f \left( 0 \right) = - 12.

a

Determine the value of k.

b

Hence find f \left( x \right).

7

Consider the gradient function \dfrac{d y}{d x} = 8 \cos 2 x.

a

Find \int 8 \cos 2x \ dx.

b

Hence find y, if y = 3 when x = \dfrac{\pi}{4}.