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VCE 12 Methods 2023

7.02 Anti-differentiation and exponential functions

Worksheet
Anti-derivatives of exponential functions
1

Find the primitive function of the following:

a

f \rq (x) = e^{ 2 x}

b

f \rq (x) = e^{ - 5 x }

c

f \rq (x) = e^{ 0.25 x}

d

f \rq (x) = 9 e^{ 3 x}

e

f \rq (x) = e^{x} + e^{ - 3 x }

f

f \rq (x) = e^{ 3 x} - e^{ - 8 x }

g

f \rq (x) = \dfrac{1}{6} \left(e^{ 0.5 x} + e^{ 3 x}\right)

h

f \rq (x) = e^{ 2 x} + \sqrt{x}

2

Find the following indefinite integrals:

a

\int e^{ - x } dx

b

\int e^{ 2 x} dx

c

\int e^{ \frac{1}{2} x} dx

d

\int 4 e^{ 2 x} dx

e

\int e^{4 - 3 x} dx

f

\int e^{3 - 2 x} dx

g

\int 4 e^{1 - 2 x} dx

h

\int \left(e^{ 0.5 x} + e^{ 3 x}\right) dx

i

\int \left(e^{t} - 5\right) dt

j

\int \left(e^{ 4 t} + t^{2}\right) dt

k

\int \left(e^{ 3 v} + v^{4}\right) dv

3

Consider the function y = x^{2} e^{x}.

a

Find an expression for \dfrac{dy}{dx}.

b

Hence find \int x \left( 2+ x \right) e^{x} dx.

Find value of C
4

A curve has gradient function \dfrac{d y}{d x} = e^{ 3 x}. Find the equation of the curve if it passes through the point \left( 0, -1 \right).

5

Consider gradient function \dfrac{d y}{d x} = 8 e^{ 4 x} - 5.

a

Find \int (8 e^{ 4 x} - 5)dx.

b

If y = 7 when x = 0, find y in terms of x.

6

Consider f \rq \left( x \right) = e^{ - 2 x } + 6 \sqrt{x}.

a

Find \int \left(e^{ - 2 x } + 6 \sqrt{x}\right)dx.

b

If f \left( 1 \right) = 3, find f \left( x \right).

7

Consider f' \left( x \right) = \dfrac{3 e^{ 2 x} + 1}{e^{x}}.

a

Find \int \left(\dfrac{3 e^{ 2 x} + 1}{e^{x}}\right)dx.

b

If f \left( 0 \right) = 4, find f \left( x \right).

8

Find the equation of the curve given the gradient function and a point on the curve:

a

\dfrac{d y}{d x} = e^{ 2 x}, and point \left(0, 6 \right).

b

\dfrac{d y}{d x} = e^{ x} -2x, and point \left(0, 4 \right).

9

A curve has gradient function \dfrac{d y}{d x} = e^{ k x} for some constant k. The point \left(1, e^{3}\right) lies both on the gradient function and also the original curve y.

a

Determine the value of k.

b

Find y in terms of x.

10

Find y in terms of x given the following information:

  • \dfrac{d y}{d x} = k e^{x} + 2, for some constant k

  • When x = 0: \, y = - 1, and \dfrac{d y}{d x} = 5

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Outcomes

U34.AoS3.6

anti-derivatives of polynomial functions and functions of the form f(ax+b) where f is x^n, e^x, sin(x), cos(x) and linear combinations of these

U34.AoS3.19

find anti-derivatives of polynomial functions and power functions, functions of the form f(ax+b) where f is x^n, e^x, sine or cosine, and simple linear combinations of these, using pattern recognition, or by hand

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