Anti-differentiation

Recognise the indefinite integral as the reverse of differentiation; apply the standard results for

x^n

e^{ax+b}

\frac{1}{x}

\sin x

\cos x

and

\sec^2 x

; use the reverse chain rule.

Worked examples

Using the power rule and an initial condition

Straightforward

Problem

A function

F(x)

satisfies

F'(x) = 4x^3 - 6x

and

F(1) = 2

. Find

F(x)

Integrating exponential and reciprocal functions

Moderate

Problem

Find

\displaystyle\int \!\left(e^{5x} + \frac{4}{x}\right)dx

for

x > 0

Reverse chain rule

Challenging

Problem

Find

\displaystyle\int 6x^2(x^3 - 1)^4\,dx

Practise

Open Math

Anti-differentiation

Calculus · MAV-12-05

Name:

Date:

Q1Straightforward

Which expression is equal to

\displaystyle\int x^4\,dx

A. $4x^3 + C$
B. $\dfrac{x^5}{5} + C$
C. $\dfrac{x^5}{4} + C$
D. $x^5 + C$

Q2Straightforward

Which expression is equal to

\displaystyle\int e^x\,dx

A. $xe^{x-1} + C$
B. $\ln x + C$
C. $e^x + C$
D. $e^{x+1} + C$

Q3Straightforward

Which expression is equal to

\displaystyle\int \frac{1}{x}\,dx

for

x > 0

A. $-\dfrac{1}{x^2} + C$
B. $\ln x + C$
C. $\dfrac{1}{x^2} + C$
D. $e^x + C$

Q4Straightforward

A function

F(x)

satisfies

F'(x) = 3x^2

and

F(0) = 4

. Find

F(2)

Q5Straightforward

A function

F(x)

satisfies

F'(x) = \cos x

and

F(0) = 2

. Find

F\!\left(\dfrac{\pi}{2}\right)

Q6Moderate

Which expression is equal to

\displaystyle\int e^{3x}\,dx

A. $3e^{3x} + C$
B. $e^{3x} + C$
C. $\dfrac{e^{3x}}{3} + C$
D. $\dfrac{e^{3x+1}}{3x+1} + C$

Q7Moderate

A function

F(x)

satisfies

F'(x) = e^{2x}

and

F(0) = \dfrac{1}{2}

. Find

F(1)

, correct to two decimal places.

Q8Moderate

A function

F(x)

satisfies

F'(x) = \sin(3x)

and

F(0) = \dfrac{1}{3}

. Find

F\!\left(\dfrac{\pi}{6}\right)

. Give your answer as a fraction.

Q9Moderate

Which expression is equal to

\displaystyle\int \sec^2 x\,dx

A. $\sec x \tan x + C$
B. $2\sec x + C$
C. $\tan x + C$
D. $\sin^2 x + C$

Q10Moderate

A function

F(x)

satisfies

F'(x) = \dfrac{3}{x}

for

x > 0

, and

F(1) = 0

. Find

F(e^2)

Q11Challenging

Using the reverse chain rule, which expression is equal to

\displaystyle\int 2x(x^2 + 1)^5\,dx

A. $10x(x^2+1)^4 + C$
B. $\dfrac{(x^2+1)^6}{6} + C$
C. $\dfrac{(x^2+1)^6}{3} + C$
D. $5(x^2+1)^4 + C$

Q12Challenging

A function

F(x)

satisfies

F'(x) = 6x^2 - 4x + 1

and

F(1) = 3

. Find

F(-1)

Q13Challenging

A function

F(x)

satisfies

F'(x) = e^{-2x}

and

F(0) = 0

. Find

F(1)

, correct to two decimal places.

Worked solutions and answers at openmath.au/year-12/advanced/integral-calculus/anti-differentiation