Topics
7. Introduction to Integral Calculus
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AustraliaNSW
Year 12

7.01 Anti-differentiation and integration

Lesson
Worksheet
Practice
Worksheet
Integration of power functions
1

Determine the primitive of the following:

a
x^{5}
b
9 x^{2}
c
x^{ - 6 }
d
10 x + 7
e
9 x^{2} + 4 x - 6
f
15 x^{4} + 16 x^{3}
g
3x^2+6x-10
h
8x^5 - 12x^-2
2

Integrate the following to find f(x):

a
f' \left( x \right) = 8 x
b
f' \left( x \right) = 16x^3-7x^2+8x
c
f' \left( x \right) = 9
d
f' \left( x \right) = 5x^4 - 2x^{\frac{1}{2}}
e
f' \left( x \right) = 4x^3 + 6x^2 - 7x + 1
f
f' \left( x \right) = 10x^4 - 12x^3 + 6x
3

Integrate the following to find F(x):

a
f \left( x \right) = \dfrac{x^{6}}{4} + \dfrac{x^{2}}{3}
b
f \left( x \right) = 4 x^{\frac{2}{5}} + 3 x^{\frac{4}{7}}
c
f \left( x \right) = x^{ - \frac{3}{7} } + x^{ - \frac{2}{5} }
d
f \left( x \right) = 8 x^{3} + 3 x^{\frac{5}{3}} - 3
4

Integrate the following to find an expression for y:

a
\dfrac{dy}{dx} = \left( 5 x - 2\right) \left( 3 x - 4\right)
b
\dfrac{dy}{dx} = \left(x + 4\right) \left(x + 6\right)
c
\dfrac{d y}{d x} = \dfrac{15}{x^{6}}
d
\dfrac{d y}{d x} = \dfrac{10}{x^{6}} - \dfrac{9}{x^{4}}
e
\dfrac{d y}{d x} = 18 \sqrt{x}
f
\dfrac{d y}{d x} = \sqrt{x}
g
\dfrac{d y}{d x} = \dfrac{6}{\sqrt{x}}
h
y' = x^{2} \left( 10 x^{2} - 9 x\right)
i
y' = x \sqrt{x}
j
y' = \dfrac{x^{3} + 4}{x^{3}}
k
y' = \dfrac{x^{2}-8x}{x}
l
y' = \dfrac{2x^{5} - 6x^4 + 10x^2}{2x^{2}}
5

Evaluate the following indefinite integrals:

a
\int 4 \,dx
b
\int x^{3} \,dx
c
\int 4 x^{3} \,dx
d
\int x^{5} \,dx
e
\int x^{ - 5 } \,dx
f
\int \dfrac{1}{4} x^{2} \,dx
g
\int x^{\frac{1}{4}} \,dx
h
\int x^{\frac{4}{5}} \,dx
i
\int 5 x^{\frac{3}{4}} \,dx
j
\int \left(x^{2} + 4 x\right) \,dx
k
\int \left( 3 x^{ - 2 } - 8 x\right) \,dx
l
\int \sqrt{x} \,dx
m
\int \left(x^{\frac{4}{5}} + 5 x^{\frac{2}{3}}\right) \,dx
n
\int \left(3 x^{2} + 6 x + 3\right) \,dx
o
\int \left(4 x^{3} + 3 x^{2}\right) \,dx
p
\int \left(3 x + 3 x^{2} + x^{3}\right) \,dx
q
\int \dfrac{5}{x^{2}} \,dx
r
\int \left(5 x^{\frac{7}{3}} + \dfrac{9}{15} x^{\frac{5}{4}}\right) \,dx
s
\int \left(\dfrac{x^{2}}{5} + \dfrac{x^{3}}{4} + 3\right) \,dx
t
\int \left(\dfrac{x^{3}}{3} - \dfrac{3}{x^{3}}\right) \,dx
u
\int \left(5 - 3 t - 4 t^{2}\right) \, dt
v
\int a y^{3} \, dy
Applications
6

For each of the following gradient functions:

i

State what type of function the antiderivative, f\left(x\right), is.

ii

Find f(x).

iii

Sketch a possible graph for the antiderivative f\left(x\right).

a
f\rq\left(x\right)=6
b
f'\left(x\right) =- 8 x
c
f\rq\left(x\right)=2
d
f'\left(x\right)= 4 x + 3
e
f'\left(x\right)=8 x + 8
f
f'\left(x\right)=7 x^{2}
g
f'\left(x\right)=- 5 x \left(x + 2\right)
h
f'\left(x\right)= - 7 x^{2}
7

Consider the gradient function f'\left(x\right)=\dfrac{3}{x^{2}}.

a

Find f\left(x\right).

b

Sketch a possible graph of the antiderivative f\left(x\right).

8

The gradient function, f'\left(x\right), has only one x-intercept at \left( - 4 , 0\right), a y-intercept at \left(0, - 3 \right) and a constant gradient.

a

Find f'\left(x\right).

b

Find f\left(x\right).

c

Sketch a possible graph of the antiderivative f\left(x\right).

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MA12-7

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

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