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9. Further Integral Calculus
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AustraliaNSW
Year 12

9.04 Further integration problems

Lesson
Worksheet
Practice
Worksheet
Further integration
1

Find the primitive of the following:

a
\dfrac{\sin x}{1 - \cos x}
b
6 \sin x - \cos x
c
e^{ 2 x} + \sqrt{x}
2

Find the following indefinite integrals:

a
\int \left(e^{ 4 t} + t^{2}\right) dt
b
\int \left(e^{ 3 v} + v^{4}\right) dv
c
\int \left(\cos x - \sin 3 x\right) dx
d
\int \dfrac{3 \sin 3 x}{\cos 3 x} dx
e
\int \dfrac{3 \cos x}{5 - \sin x} dx
f
\int \dfrac{\left(x - 4\right)^{2}}{x^{2}} dx
g
\int \dfrac{\sqrt{x} + 3}{x} dx
h
\int \left(x^{\frac{1}{4}} + \sin 3 x\right) dx
i
\int \cos x \left(4 + \tan x\right) dx
j
\int \left(\sin \left(\dfrac{x}{3}\right) + \cos \left(\dfrac{x}{3}\right)\right) dx
3

Find the exact value of the following definite integrals:

a
\int_{ - 4 }^{4} \left(e^{x} + e^{ 2 x}\right)^{2} dx
b
\int_{3}^{4} \left(e^{ 2 x} + 4\right)^{2} dx
c
\int_{ - 4 }^{3} \dfrac{e^{ - 2 x} + e^{ 3 x}}{e^{x}} dx
d
\int_{1}^{16} \left(e^{ 4 x} + x^{2} - \sqrt{x}\right) dx
e
\int_{\ln 2}^{\ln 6} \dfrac{e^{ 3 x} + 9}{e^{x}} dx
f
\int_{2}^{5} \left( 6 x^{2} + \dfrac{1}{x}\right) dx
g
\int_{4}^{8} \left(\dfrac{3}{x} + \dfrac{1}{x^{2}}\right) dx
h
\int_{2}^{7} \left(x - \dfrac{1}{x^{2}}\right)^{2} dx
i
\int_{\frac{\pi}{3}}^{\frac{2 \pi}{3}} \dfrac{\sin x}{2 + \cos x} dx
j
\int_{0}^{\frac{\pi}{6}} \left(\sin x + \cos x\right) dx
k
\int_{ - \frac{\pi}{6} }^{\frac{\pi}{6}} \left( 4 \cos x + \cos 4 x\right) dx
l
\int_{0}^{\frac{\pi}{6}} 6 \left(\tan ^{2}\left( 2 x\right) - 1\right) dx
m
\int_{ - \frac{\pi}{2} }^{\frac{\pi}{2}} \left( 8 \cos \left( 2 x\right) - 9 \sin \left( 3 x\right)\right) dx
n
\int_{\frac{4 \pi}{3}}^{\frac{5 \pi}{4}} \left( 6 \sec ^{2}\left(x\right) + 3\right) dx
o
\int_{ - \pi }^{\pi} \left(\sin \left(\dfrac{\theta}{6}\right) - \cos \left(\dfrac{\theta}{6}\right)\right) d\theta
p
\int_{0}^{\frac{\pi}{12}} \left(\sec ^{2}\left( 3 x\right) + 8 x\right) dx
q
\int_{0}^{\frac{\pi}{3}} \left( 2 \cos 3 x + \dfrac{1}{4} \sin 2 x\right) dx
4

Consider the function y = x \ln x.

a

Differentiate the function.

b

Evaluate \int_{2}^{3} \ln x \, dx, correct to two decimal places.

5

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.

6

Consider the function y = e^{x} + e^{ - x }.

a

Differentiate the function.

b

Hence find the exact value of \int_{0}^{3} \dfrac{e^{x} - e^{ - x }}{e^{x} + e^{ - x }} dx.

7

Consider the function y = e^{ 3 x} \left(x - \dfrac{1}{3}\right).

a

Find y'.

b

Hence find the exact value of \int_{6}^{9} x e^{ 3 x} dx.

Area bounded by curves
8

Calculate the exact area bounded by:

a

The curve y = e^{\frac{x}{4}} + e^{ - \frac{x}{4} }, the coordinate axes, and the line x = \dfrac{1}{2}.

b

The curve y = 3 \left(\sin x + 1\right), the x-axis and the lines x = 0 and x = 2 \pi.

c

The curve f \left( x \right) = x + \dfrac{5}{x + 5}, the x-axis and the lines x = 0 and x = 2.

d

The curve y = \dfrac{\cos x}{1 + \sin x}, the x-axis and the lines x = 0 and x = \dfrac{\pi}{6}.

e

The curve y = 2 + \dfrac{1}{4} \sin 2 x, the x-axis and the lines x = 0 and x = \dfrac{\pi}{2}.

f

The curve y = \dfrac{1}{3} + 3 \cos x, the x-axis and the lines x = - \dfrac{\pi}{6} to x = \dfrac{\pi}{6}.

g

The curve y = e^{1 - x}, the line y = x and the line x = 3.

h

The curves y = e^{x} and y = e^{ - x }, and the line x = 3.

i

The curves y = e^{ 5 x} and y = e^{ - 5 x }, and the lines x = - 2 and x = 2.

j

The curves y = \sin 2 x and y = \cos x over the domain \dfrac{\pi}{6} \leq x \leq \dfrac{\pi}{2}.

k

The curves y = e^{ 2 x} and y = e^{ - x }, the x-axis, and the lines x = - 2 and x = 2.

9

Consider the integral \int_{0}^{\frac{\pi}{2}} \left(2 - \cos x\right) dx.

a

Sketch the graph of y = 2 - \cos x, clearly showing the area represented by \int_{0}^{\frac{\pi}{2}} \left(2 - \cos x\right) dx.

b

Hence find the exact area represented by the integral \int_{0}^{\frac{\pi}{2}} \left(2 - \cos x\right) dx.

10

Consider the functions y = \left|x\right| and y = \dfrac{2}{x}.

a

Sketch the graphs of these functions on the same number plane.

b

Find the exact x-value of the point of intersection of the two functions.

c

Calculate the exact area between the two curves, the x-axis and the line x = 5.

11

Consider the functions y = - x^{2} and y = \dfrac{1}{4 - x} for 0 \leq x \leq 3.

a

Sketch the graphs of these functions on the same number plane.

b

Calculate the exact area bound by the curves between x = 1 and x = 3.

12

Consider the function y = x + \dfrac{1}{x}.

a

State the domain of the function.

b

State the value the function approaches as x \to \infty.

c

Calculate the exact area enclosed by the function, the x-axis, and the lines x = 2 and x = 12.

13

Consider the functions f \left( x \right) = e^{x} + 12 and g \left( x \right) = 4 x + e^{3}.

a

Find the x-coordinate of the point of intersection that is to the right of the origin.

b

Calculate the exact area in the first quadrant bounded by the two curves and the y-axis.

14

Consider the functions y = e^{\frac{x}{6}} and y = e^{\frac{3}{2}}.

a

Find the x-coordinate of the point of intersection.

b

Calculate the exact area bounded by y = e^{\frac{x}{6}}, the y-axis and the line y = e^{\frac{3}{2}}.

15

Consider the functions f \left( x \right) = 3 - e^{x} and g \left( x \right) = e^{x} + 1 .

a

Find the x-coordinate of the point of intersection.

b

Calculate the exact area bound between the two curves and the line x = \ln 3.

Applications
16

Given that \int_{3}^{5} \dfrac{e^{x}}{1 + e^{x}} dx = \ln c, find the value of c.

17

Consider the graph of the functions y = x^{2} and y = \dfrac{1}{x}:

Solve for the exact value of k such that: \int_{1}^{k} \dfrac{1}{x} \, dx = \int_{0}^{1} x^2 \, dx

-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
18
a

Prove that \left(1 + \tan ^{2}x\right) \cos ^{2}x = 1.

b

Hence, find the exact value of \int_{\frac{\pi}{8}}^{\frac{\pi}{5}} \left(1 + \tan ^{2}x\right) \cos ^{2}x dx.

19

Find the function h \left( x \right), given that h(0) = \ln 8 and h' \left( x \right)=\dfrac{\cos x \sin x}{\sin ^{2}\left(x\right) + 8}.

20

Consider the function f \left( x \right) = e^{ 3 x} - e^{ - 3 x }.

a

Find the x-intercept of the function.

b

Find the limiting function as x approaches +\infty.

c

Show that the function is odd.

d

Find the exact area bounded by the curve, the coordinate axes, and the line x = 3.

e

Find the exact area bound by the curve, the x-axis, and the lines x = 1 and x = - 1.

21

The cross-section of a satellite dish can be estimated by the area bound by the hyperbola y = \dfrac{32}{x} \,and the line y = - x + 12 \,, as shown in the diagram:

a

Find the x-values of the points of intersection of the hyperbola and the line.

b

Hence determine the cross-sectional area of the satellite dish, correct to two decimal places.

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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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