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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.

Worked examples

Deriving a reduction formula

Straightforward

Problem

Let In=∫0π/2sin⁡nx dxI_n = \displaystyle\int_0^{\pi/2} \sin^n x\,dx. Derive the reduction formula In=n−1n In−2I_n = \dfrac{n-1}{n}\,I_{n-2}.

Applying a reduction formula

Moderate

Problem

Evaluate ∫0π/2sin⁡5x dx\displaystyle\int_0^{\pi/2} \sin^5 x\,dx using In=n−1n In−2I_n = \dfrac{n-1}{n}\,I_{n-2} with I1=1I_1 = 1.

Choosing the right technique

Challenging

Problem

Evaluate ∫1e(ln⁡x)3x dx\displaystyle\int_1^e \frac{(\ln x)^3}{x}\,dx.

Practise

Q1·Straightforward
Let In=∫0π/2sin⁡nx dxI_n = \displaystyle\int_0^{\pi/2} \sin^n x\,dx. The reduction formula is In=n−1n In−2I_n = \dfrac{n-1}{n}\,I_{n-2}, with I0=π2I_0 = \dfrac{\pi}{2} and I1=1I_1 = 1.

Use this formula to find I4I_4. Give your answer to two decimal places.
Q2·Straightforward
Let In=∫01xnex dxI_n = \displaystyle\int_0^1 x^n e^x\,dx. The reduction formula is In=e−n In−1I_n = e - n\,I_{n-1}, with I0=e−1I_0 = e - 1.

Find I2I_2. Give your answer to two decimal places.
Q3·Straightforward
Let In=∫0π/4tan⁡nx dxI_n = \displaystyle\int_0^{\pi/4} \tan^n x\,dx. The reduction formula is In+In−2=1n−1I_n + I_{n-2} = \dfrac{1}{n-1} (for n≥2n \ge 2).

Given that I1=ln⁡22I_1 = \dfrac{\ln 2}{2}, find I3I_3. Give your answer to two decimal places.
Q4·Moderate
Let In=∫0π/2cos⁡nx dxI_n = \displaystyle\int_0^{\pi/2} \cos^n x\,dx for integer n≥2n \ge 2.

Write cos⁡nx=cos⁡n−1x⋅cos⁡x\cos^n x = \cos^{n-1}x \cdot \cos x and apply integration by parts (let u=cos⁡n−1xu = \cos^{n-1}x and dv=cos⁡x dxdv = \cos x\,dx) to show that
In=n−1n In−2.I_n = \frac{n-1}{n}\,I_{n-2}.

✎ Work this one through on paper — proofs are self-assessed.

Q5·Moderate
Let In=∫0π/2sin⁡nx dxI_n = \displaystyle\int_0^{\pi/2} \sin^n x\,dx, with reduction formula In=n−1n In−2I_n = \dfrac{n-1}{n}\,I_{n-2} and I1=1I_1 = 1.

Use the formula to find I5I_5. Express your answer as a fraction.
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Q6·Moderate
Evaluate ∫01x(x+1)2 dx\displaystyle\int_0^1 \frac{x}{(x+1)^2}\,dx using the substitution u=x+1u = x + 1. Give your answer to two decimal places.
Q7·Moderate
Evaluate ∫01x2ln⁡x dx\displaystyle\int_0^1 x^2 \ln x\,dx using integration by parts with u=ln⁡xu = \ln x and dv=x2 dxdv = x^2\,dx. Give your answer to two decimal places.

(Note: lim⁡x→0+x3ln⁡x=0\lim_{x \to 0^+} x^3 \ln x = 0.)
Q8·Moderate
Evaluate ∫0πxsin⁡x dx\displaystyle\int_0^{\pi} x \sin x\,dx using integration by parts. Give your answer to two decimal places.
Q9·Moderate
Show that ∫0π/2sin⁡2xcos⁡2x dx=π16\displaystyle\int_0^{\pi/2} \sin^2 x \cos^2 x\,dx = \dfrac{\pi}{16}.

(Hint: use the identity sin⁡xcos⁡x=12sin⁡2x\sin x\cos x = \tfrac{1}{2}\sin 2x, then the double-angle identity sin⁡2θ=1−cos⁡2θ2\sin^2 \theta = \tfrac{1-\cos 2\theta}{2}.)

✎ Work this one through on paper — proofs are self-assessed.

Q10·Challenging
Use the reduction formula In=n−1n In−2I_n = \dfrac{n-1}{n}\,I_{n-2} with I0=π2I_0 = \dfrac{\pi}{2} to evaluate I6=∫0π/2sin⁡6x dxI_6 = \displaystyle\int_0^{\pi/2} \sin^6 x\,dx. Give your answer to two decimal places.
Q11·Challenging
Evaluate ∫013(1+x)(1+x2) dx\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 A1+x+Bx+C1+x2\dfrac{A}{1+x} + \dfrac{Bx + C}{1+x^2} and find the constants.)
Q12·Challenging
Let Jn=∫0π/2sin⁡nx dxJ_n = \displaystyle\int_0^{\pi/2} \sin^n x\,dx with J0=π2J_0 = \dfrac{\pi}{2} and J1=1J_1 = 1.

Show, by repeated application of the reduction formula Jn=n−1nJn−2J_n = \dfrac{n-1}{n}J_{n-2}, that
J6=5π32J_6 = \frac{5\pi}{32}

and verify this satisfies the pattern J2k=(2k−1)!!(2k)!!⋅π2J_{2k} = \dfrac{(2k-1)!!}{(2k)!!}\cdot\dfrac{\pi}{2} for k=3k=3.

✎ Work this one through on paper — proofs are self-assessed.