Further integration

Extend integration techniques to include integration by parts, partial fractions (including repeated and irreducible quadratic factors), the t-substitution, trigonometric integrals, reduction formulae, and problems requiring a strategic choice of method.

Calculus · MEX-12-03

Sub-topics

Integration techniques

Integrate using integration by parts (

\int u\,dv = uv - \int v\,du

); decompose rational functions using partial fractions including repeated and irreducible quadratic factors; apply the

t = \tan(x/2)

Weierstrass substitution; evaluate trigonometric integrals and powers.

Recurrence and harder integrals

Derive and apply reduction formulae to evaluate families of integrals; select and combine integration techniques for definite integrals that require a strategic approach.

Open Math

Further integration

Calculus · MEX-12-03

Name:

Date:

Integration techniques

Q1Straightforward

Use integration by parts to evaluate

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

. (Let

u = x

and

dv = e^x\,dx

Q2Straightforward

Use integration by parts to evaluate

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

. Give your answer to two decimal places.

Q3Straightforward

Evaluate

\displaystyle\int_1^e \ln x\,dx

using integration by parts. (Let

u = \ln x

and

dv = dx

Q4Straightforward

Evaluate

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

by applying integration by parts twice. Give your answer to two decimal places.

Q5Moderate

Use partial fractions to evaluate

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

. Give your answer to two decimal places.

Q6Moderate

Evaluate

\displaystyle\int_2^3 \frac{2x+1}{x^2+x}\,dx

. Give your answer to two decimal places.

Q7Moderate

Evaluate

\displaystyle\int_0^{\pi/4} \tan^2 x\,dx

using the identity

\tan^2 x = \sec^2 x - 1

. Give your answer to two decimal places.

Q8Moderate

Evaluate

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

using the double-angle identity

\sin^2 x = \dfrac{1 - \cos 2x}{2}

. Give your answer to two decimal places.

Q9Moderate

Use partial fractions to evaluate

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

. Give your answer to two decimal places.

Q10Challenging

Use the Weierstrass substitution

t = \tan\!\left(\dfrac{x}{2}\right)

to evaluate

\displaystyle\int_0^{\pi/2} \frac{1}{1 + \sin x}\,dx

.

Recall:

\sin x = \dfrac{2t}{1+t^2}

and

dx = \dfrac{2}{1+t^2}\,dt

Q11Challenging

Evaluate

\displaystyle\int_0^{\pi/2} \cos^3 x\,dx

by writing

\cos^3 x = \cos x(1 - \sin^2 x)

and using the substitution

u = \sin x

. Express your answer as a fraction.

Q12Challenging

Evaluate

\displaystyle\int_0^{\pi} e^x \sin x\,dx

using integration by parts twice. (Hint: after two applications, you will find the original integral appears on both sides of the equation.) Give your answer to two decimal places.

Recurrence and harder integrals

Q13Straightforward

Let

I_n = \displaystyle\int_0^{\pi/2} \sin^n x\,dx

. The reduction formula is

I_n = \dfrac{n-1}{n}\,I_{n-2}

, with

I_0 = \dfrac{\pi}{2}

and

I_1 = 1

.

Use this formula to find

I_4

. Give your answer to two decimal places.

Q14Straightforward

Let

I_n = \displaystyle\int_0^1 x^n e^x\,dx

. The reduction formula is

I_n = e - n\,I_{n-1}

, with

I_0 = e - 1

.

Find

I_2

. Give your answer to two decimal places.

Q15Straightforward

Let

I_n = \displaystyle\int_0^{\pi/4} \tan^n x\,dx

. The reduction formula is

I_n + I_{n-2} = \dfrac{1}{n-1}

(for

n \ge 2

).

Given that

I_1 = \dfrac{\ln 2}{2}

, find

I_3

. Give your answer to two decimal places.

Q16Moderate

Let

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

for integer

n \ge 2

.

Write

\cos^n x = \cos^{n-1}x \cdot \cos x

and apply integration by parts (let

u = \cos^{n-1}x

and

dv = \cos x\,dx

) to show that

I_n = \frac{n-1}{n}\,I_{n-2}.

Q17Moderate

Let

I_n = \displaystyle\int_0^{\pi/2} \sin^n x\,dx

, with reduction formula

I_n = \dfrac{n-1}{n}\,I_{n-2}

and

I_1 = 1

.

Use the formula to find

I_5

. Express your answer as a fraction.

Q18Moderate

Evaluate

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

using the substitution

u = x + 1

. Give your answer to two decimal places.

Q19Moderate

Evaluate

\displaystyle\int_0^1 x^2 \ln x\,dx

using integration by parts with

u = \ln x

and

dv = x^2\,dx

. Give your answer to two decimal places.

(Note:

\lim_{x \to 0^+} x^3 \ln x = 0

Q20Moderate

Evaluate

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

using integration by parts. Give your answer to two decimal places.

Q21Moderate

Show that

\displaystyle\int_0^{\pi/2} \sin^2 x \cos^2 x\,dx = \dfrac{\pi}{16}

.

(Hint: use the identity

\sin x\cos x = \tfrac{1}{2}\sin 2x

, then the double-angle identity

\sin^2 \theta = \tfrac{1-\cos 2\theta}{2}

Q22Challenging

Use the reduction formula

I_n = \dfrac{n-1}{n}\,I_{n-2}

with

I_0 = \dfrac{\pi}{2}

to evaluate

I_6 = \displaystyle\int_0^{\pi/2} \sin^6 x\,dx

. Give your answer to two decimal places.

Q23Challenging

Evaluate

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

using partial fractions. Give your answer to two decimal places.

(Express the integrand as

\dfrac{A}{1+x} + \dfrac{Bx + C}{1+x^2}

and find the constants.)

Q24Challenging

Let

J_n = \displaystyle\int_0^{\pi/2} \sin^n x\,dx

with

J_0 = \dfrac{\pi}{2}

and

J_1 = 1

.

Show, by repeated application of the reduction formula

J_n = \dfrac{n-1}{n}J_{n-2}

, that

J_6 = \frac{5\pi}{32}

and verify this satisfies the pattern

J_{2k} = \dfrac{(2k-1)!!}{(2k)!!}\cdot\dfrac{\pi}{2}

for

k=3

Worked solutions and answers at openmath.au/year-12/extension-2/further-integration