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9. Further Integral Calculus
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AustraliaNSW
Year 12

9.03 Integrating trigonometric functions

Lesson
Worksheet
Practice
Worksheet
Trigonometric functions
1

Find the primitive of the following:

a

7 \sin x

b

- 7 \cos x

c

\sin 7 x

d

\cos 6 x

e

9 \sin 3 x

f

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

g

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

h
\dfrac{\cos x}{\sin x}
i
4 \sec ^{2}\left(x\right)
j
\cos \left(x + \dfrac{\pi}{6}\right)
k
\sin \left(x - \dfrac{\pi}{6}\right)
l
\sin \left( 5 x - \dfrac{\pi}{4}\right)
m
- 8 \sin \left( 2 x + \dfrac{\pi}{6}\right)
n
\cos \left(\dfrac{x}{8} + \dfrac{\pi}{3}\right)
o
8 \cos \left( 4 x - \dfrac{\pi}{3}\right)
p
2 \sin \left(\dfrac{x}{7} - \dfrac{\pi}{4}\right)
q
- 4 \cos \left(\dfrac{x}{7} + \dfrac{\pi}{4}\right)
2

Find the following indefinite integrals:

a
\int \sec ^{2}\left(x\right) \, dx
b
\int \sec ^{2}\left( 3 x\right) \, dx
c
\int 3 \sec ^{2}\left(\dfrac{x}{4}\right) \, dx
d
\int \tan x \, dx
e
\int \tan ^{2}\left(x\right) dx
f
\int \sin\left( 2x + \pi\right) \, dx
g
\int \sec^2 \left(2x\right)\, dx
h
\int \cos\left( 2x + \dfrac{\pi}{2}\right) \, dx
i
\int \left(x^3+\sin 2x \right)\, dx
j
\int \left(\sin \dfrac{x}{2} + \cos \dfrac{x}{2}\right) \, dx
k
\int \left(\cos \dfrac{x}{2} + \dfrac{1}{3} \sin3x\right) \, dx
3

Find the exact value of the following definite integrals:

a
\int_{\frac{3 \pi}{2}}^{ 2 \pi} \cos x \, dx
b
\int_{\frac{\pi}{2}}^{\pi} \left( - \sin x \right) \, dx
c
\int_{\frac{\pi}{6}}^{\frac{7 \pi}{6}} 2 \sin x \, dx
d
\int_{\frac{2\pi}{3}}^{\pi} \sec^2\left(x\right)\, dx
e
\int_{\frac{\pi}{6}}^{\frac{7 \pi}{6}} 2 \cos x \, dx
f
\int_{0}^{\frac{\pi}{3}} 3 \sec ^{2}\left(x\right) \, dx
g
\int_{0}^{\pi} \sec ^{2}\left(\dfrac{x}{3}\right) \, dx
h
\int_{ - \frac{\pi}{12} }^{\frac{\pi}{12}} \sec ^{2}\left( 3 x\right) \, dx
i
\int_{\frac{\pi}{4}}^{\frac{5 \pi}{18}} \sec ^{2}\left( 3 x\right) \, dx
j
\int_{ - \frac{\pi}{6} }^{\frac{7 \pi}{6}} \sin 3 x \, dx
k
\int_{ - \frac{\pi}{6} }^{\frac{7 \pi}{6}} \cos 3 x \, dx
l
\int_{ - \frac{2 \pi}{3} }^{ 2 \pi} \sin \left(\dfrac{x}{4}\right) \, dx
m
\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \sin \left(x + \pi\right) \, dx
n
\int_{ - \frac{4 \pi}{3} }^{\frac{2 \pi}{3}} \cos \left(\dfrac{x}{4}\right) \, dx
o
\int_{ - 6 }^{9} \sin \left(\dfrac{\pi x}{3}\right) \, dx
p
\int_{ - 8 }^{4} \cos \left(\frac{\pi x}{2}\right) \, dx
q
\int_{ - \frac{\pi}{2} }^{\pi} \cos \left( 4 x - \dfrac{\pi}{2}\right) \, dx
r
\int_0^\frac{\pi}{2} \sec^2 \left(2x\right)\, dx
s
\int_\frac{\pi}{2}^\pi \cos \dfrac{x}{2}\, dx
t
\int_\frac{\pi}{6}^\frac{\pi}{4} \left(\cos 2x -\sin x\right) \, dx
u
\int_\frac{\pi}{6}^\frac{\pi}{2} \left(\sin 2x -\cos 2x\right) \, dx
v
\int_0^{\pi} \left(\sin \dfrac{x}{3} + \cos \dfrac{x}{3}\right) \, dx
w
\int_0^\frac{\pi}{3} \tan x \, dx
x
\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \cot x\, dx
y
\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \dfrac{3 - \cos ^{3}\left(x\right)}{\cos ^{2}\left(x\right)} \, dx
z
\int_0^\frac{\pi}{2} \left(4\cos 4x -\dfrac{\sin 2x}{2}\right) \, dx
4

Consider the function f(x)=\cos ^{5}x.

a

Find \dfrac{d}{dx} \left(\cos ^{5}x\right).

b

Hence find \int_{0}^{\pi} \sin x \cos ^{4}x \, dx.

5

Consider the function y = \tan x on the interval 0 \leq x < \dfrac{\pi}{2}.

a

Rewrite the equation of the function in terms of \sin x and \cos x.

b

Hence find \int \tan x \,dx.

6

Given that \sin 3 t = 3 \sin t - 4 \sin ^{3}t, find the indefinite integral \int 3 \sin ^{3}t \ dt.

7

Use the identity \tan x = \dfrac{\sin x}{\cos x} to show that \int_0^\frac{\pi}{4} \tan x \, dx = \dfrac{1}{2}\ln 2.

8

Use the identity \cot x = \dfrac{\cos x}{\sin x} \text{ to find }\int_\frac{\pi}{6}^\frac{\pi}{2} \cot x \, dx .

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Outcomes

MA12-3

applies calculus techniques to model and solve problems

MA12-7

applies the concepts and techniques of indefinite and definite integrals in the solution of problems

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