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Deriving reduction formulae

Edexcel IAL Further Mathematics · WFM03 Further Pure Mathematics 3

A reduction formula links an integral involving a power n to the same integral with a smaller power. Apply it repeatedly and you can evaluate integrals like ∫ sin6 x dx without ever expanding.

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What a reduction formula is

Write I_n for the integral, for example I_n = ∫ from 0 to 1 of xn ex dx. A reduction formula expresses I_n in terms of I_(n-1) or I_(n-2). You then need one starting value such as I_0 or I_1, found directly.

Deriving one by parts

I_n = ∫ from 0 to 1 of xn ex dx. Take u = xn (du/dx = nxn-1) and dv/dx = ex (v = ex):

I_n = [xn ex] from 0 to 1 - n ∫ from 0 to 1 of xn-1 ex dx = e - n I_(n-1).

Choose u so that its derivative lowers the power.

Powers of sin and cos

I_n = ∫ from 0 to π/2 of sinn x dx. Write sinn x = sinn-1 x × sin x and use parts with u = sinn-1 x, v = -cos x. The boundary term vanishes, and the cos2 x that appears is replaced by 1 - sin2 x:

I_n = (n - 1)(I_(n-2) - I_n), so n I_n = (n - 1) I_(n-2) for n ≥ 2.

The same formula holds for ∫ from 0 to π/2 of cosn x dx. Starting values: I_0 = π/2, I_1 = 1.

Without parts

Sometimes an identity does the job. For I_n = ∫ from 0 to π/4 of tann x dx, write tann x = tann-2 x (sec2 x - 1):

I_n = [tann-1 x/(n - 1)] from 0 to π/4 - I_(n-2), so I_n + I_(n-2) = 1/(n - 1).

Worked example

Use n I_n = (n - 1) I_(n-2), where I_n = ∫ from 0 to π/2 of sinn x dx, to find I_4.

  1. I_4 = (3/4) I_2
  2. I_2 = (1/2) I_0 and I_0 = π/2
  3. I_4 = (3/4)(1/2)(π/2)

Answer: 3π/16

Key idea: Integrate by parts (or use an identity) to link I_n to I_(n-1) or I_(n-2), then work down to a starting value you can integrate directly.

Check yourself

  1. I_n = ∫ from 0 to 1 of xn ex dx satisfies I_n = e - n I_(n-1), and I_0 = e - 1. Find I_2, to 3 decimal places.
    Show answer

    0.718. I_1 = e - (e - 1) = 1. I_2 = e - 2(1) = e - 2 = 0.718.

  2. I_n = ∫ from 0 to π/2 of sinn x dx and n I_n = (n - 1) I_(n-2). Find I_4, to 3 decimal places.
    Show answer

    0.589. I_4 = (3/4)(1/2)(π/2) = 3π/16 = 0.589.

  3. Using the same reduction formula, find the exact value of ∫ from 0 to π/2 of sin5 x dx. Give your answer as a fraction in its simplest form.
    Show answer

    8/15. I_5 = (4/5) I_3 = (4/5)(2/3) I_1 and I_1 = 1, so I_5 = 8/15.

  4. To derive a reduction formula for ∫ xn ex dx by parts, which choice is right?
    • u = ex, dv/dx = xn
    • u = xn, dv/dx = ex
    • u = x, dv/dx = xn-1 ex
    • u = xn ex, dv/dx = 1
    Show answer

    u = xn, dv/dx = ex. Differentiating xn lowers the power to n - 1, which is what links I_n to I_(n-1).

  5. J_n = ∫ from 0 to π/2 of cosn x dx. Which reduction formula holds?
    • J_n = (n - 1) J_(n-1)
    • n J_n = (n + 1) J_(n-2)
    • J_n = J_(n-2) - 1/(n - 1)
    • n J_n = (n - 1) J_(n-2)
    Show answer

    n J_n = (n - 1) J_(n-2). The derivation mirrors the sin case: parts on cosn-1 x × cos x, then sin2 x = 1 - cos2 x.

  6. I_n = ∫ from 0 to π/4 of tann x dx satisfies I_n + I_(n-2) = 1/(n - 1), with I_0 = π/4. Find I_2, to 3 decimal places.
    Show answer

    0.215. I_2 = 1/1 - I_0 = 1 - π/4 = 0.215.

  7. Using I_n = e - n I_(n-1) and I_2 = e - 2, find I_3 = ∫ from 0 to 1 of x3 ex dx, to 3 decimal places.
    Show answer

    0.563. I_3 = e - 3(e - 2) = 6 - 2e = 0.563.

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Last updated 8 November 2026. Spotted a mistake? Tell us.