Integral calculus

Develop anti-differentiation and integration skills; evaluate definite integrals using the fundamental theorem of calculus; calculate exact areas and use the trapezoidal rule.

Calculus · MAV-12-05

Sub-topics

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.

The definite integral

Evaluate definite integrals using the fundamental theorem of calculus; interpret the definite integral as the signed area under a curve; work with regions above and below the

x

-axis.

Areas and the trapezoidal rule

Calculate the exact area between a curve and an axis; find the area between two curves after determining intersection points; approximate areas numerically using the trapezoidal rule.

Open Math

Integral calculus

Calculus · MAV-12-05

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Anti-differentiation

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.

The definite integral

Q14Straightforward

Evaluate

\displaystyle\int_0^2 3x^2\,dx

Q15Straightforward

Evaluate

\displaystyle\int_1^4 \frac{1}{x}\,dx

, correct to two decimal places.

Q16Straightforward

Evaluate

\displaystyle\int_0^{\pi/2} \cos x\,dx

Q17Straightforward

Evaluate

\displaystyle\int_{-1}^{2} (2x + 1)\,dx

Q18Straightforward

Evaluate

\displaystyle\int_0^1 e^x\,dx

, correct to two decimal places.

Q19Moderate

Evaluate

\displaystyle\int_0^1 e^{2x}\,dx

, correct to two decimal places.

Q20Moderate

Evaluate

\displaystyle\int_0^{\pi/3} \sin x\,dx

. Give your answer as a fraction.

Q21Moderate

Evaluate

\displaystyle\int_0^1 (x^3 + 2x)\,dx

. Give your answer as a fraction.

Q22Moderate

The region between

y = x^2

and the

x

-axis for

-1 \leq x \leq 1

lies entirely above the

x

-axis. Find its area. Give your answer as a fraction.

Q23Moderate

Evaluate

\displaystyle\int_{-2}^{3} (x^2 - 4)\,dx

Q24Challenging

Evaluate

\displaystyle\int_1^e \frac{2x^2 + 1}{x}\,dx

, correct to two decimal places.

Q25Challenging

Define

F(x) = \displaystyle\int_0^x (t^2 + 1)\,dt

. Find

F'(3)

Q26Challenging

Find the total area enclosed between the curve

y = x^3 - x

and the

x

-axis for

-1 \leq x \leq 1

. Give your answer as a fraction.

*Note: the curve crosses the

x

-axis within this interval, so you will need to handle each sub-region separately.*

Areas and the trapezoidal rule

Q27Straightforward

Find the exact area between the curve

y = x^2 + 1

and the

x

-axis for

0 \leq x \leq 2

. Give your answer as a fraction.

Q28Straightforward

Estimate

\displaystyle\int_0^2 x^2\,dx

using the trapezoidal rule with

n = 1

sub-interval.

Q29Straightforward

Estimate

\displaystyle\int_0^2 x^2\,dx

using the trapezoidal rule with

n = 2

sub-intervals.

Q30Straightforward

Find the exact area enclosed between

y = \cos x

and the

x

-axis for

0 \leq x \leq \dfrac{\pi}{2}

Q31Straightforward

Find the exact area of the region enclosed between

y = x

and

y = x^2

for

0 \leq x \leq 1

. Give your answer as a fraction.

Q32Moderate

Find the exact area of the region enclosed between the parabola

y = x^2

and the horizontal line

y = 4

. Give your answer as a fraction.

Q33Moderate

Use the trapezoidal rule with

n = 4

sub-intervals to approximate

\displaystyle\int_1^3 \frac{1}{x}\,dx

. Give your answer correct to two decimal places.

Q34Moderate

Find the exact area of the region enclosed between

y = x^3

and

y = x

for

0 \leq x \leq 1

. Give your answer as a fraction.

Q35Moderate

Find the exact area under

y = e^x

from

x = -1

x = 1

, correct to two decimal places.

Q36Moderate

Find the exact area of the region between

y = \sin x

and

y = \cos x

for

0 \leq x \leq \dfrac{\pi}{4}

, correct to two decimal places.

Q37Challenging

Find the area of the region enclosed between the curves

y = x^2 - 2x

and

y = x

. Give your answer as a fraction.

Q38Challenging

Use the trapezoidal rule with

n = 4

sub-intervals to estimate

\displaystyle\int_0^2 (x^2 + 1)\,dx

. By how much does this approximation exceed the exact value? Give your answer as a fraction.

Q39Challenging

Find the exact area of the region enclosed between the curves

y = x^2 - 1

and

y = 2x + 2

. Give your answer as a fraction.

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