AS & A Level · Practice questions · Integration using partial fractions and f'(x)/f(x)

Split, integrate, combine.

Six original Cambridge-style questions on integrating rational functions with logarithms, partial fractions and inverse tan, with exact answers.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Is the top the derivative of the bottom?

Check first: if the numerator is \(k f'(x)\), the answer is \(k\ln|f(x)|\).
Improper? If the top has the same degree as the bottom, start with a constant term.
Split into partial fractions; substitute roots, then compare coefficients.
Integrate each piece: \(\ln\) for linear, power rule for squared, \(\ln\) plus \(\tan^{-1}\) for \(x^2 + k^2\).
Tidy with the log laws; give exact values unless asked otherwise.

Recall

[4 marks]

Find \(\displaystyle\int \dfrac{7x-1}{(x+1)(x-3)}\,dx\).

\(7x - 1 = A(x-3) + B(x+1)\). ✓
\(x = 3\): \(20 = 4B\), \(B = 5\). \(x = -1\): \(-8 = -4A\), \(A = 2\). ✓
\(\displaystyle\int \left(\dfrac{2}{x+1} + \dfrac{5}{x-3}\right) dx = 2\ln|x+1| + 5\ln|x-3| + c\). ✓

Calculation

[3 marks]

Find the exact value of \(\displaystyle\int_0^1 \dfrac{e^{x}}{e^{x}+1}\,dx\).

The top is the derivative of the bottom, so the integral is \(\ln(e^{x}+1)\). ✓
\(\ln(e+1) - \ln 2 = \ln\dfrac{e+1}{2}\). ✓

Calculation

[6 marks]

(a) Express \(\dfrac{2x^2+2}{(x+1)(x-1)^2}\) in partial fractions.

(b) Hence show that \(\displaystyle\int_2^3 \dfrac{2x^2+2}{(x+1)(x-1)^2}\,dx = 1 + \ln\dfrac{8}{3}\).

(a) \(2x^2 + 2 = A(x-1)^2 + B(x+1)(x-1) + C(x+1)\). ✓
\(x = 1\): \(4 = 2C\), \(C = 2\). \(x = -1\): \(4 = 4A\), \(A = 1\). \(x^2\): \(2 = A + B\), \(B = 1\). ✓
\(\dfrac{1}{x+1} + \dfrac{1}{x-1} + \dfrac{2}{(x-1)^2}\). ✓
(b) \(\left[\ln(x+1) + \ln(x-1) - \dfrac{2}{x-1}\right]_2^3 = (\ln 4 + \ln 2 - 1) - (\ln 3 + 0 - 2)\). ✓
\(= \ln 8 - \ln 3 + 1 = 1 + \ln\dfrac{8}{3}\). ✓

Proof

[4 marks]

Show that \(\displaystyle\int_0^{\pi/3} \dfrac{\sin x}{1+\cos x}\,dx = \ln\dfrac{4}{3}\).

The derivative of \(1 + \cos x\) is \(-\sin x\), so \(\displaystyle\int \dfrac{\sin x}{1+\cos x}\,dx = -\ln(1+\cos x)\). ✓
Upper: \(-\ln\left(1 + \dfrac{1}{2}\right) = -\ln\dfrac{3}{2}\). Lower: \(-\ln 2\). ✓
\(-\ln\dfrac{3}{2} + \ln 2 = \ln\dfrac{2}{3/2} = \ln\dfrac{4}{3}\). ✓

Calculation

[6 marks]

(a) Express \(\dfrac{x^2+2}{(x-1)(x+2)}\) in the form \(1 + \dfrac{A}{x-1} + \dfrac{B}{x+2}\).

(b) Hence find the exact value of \(\displaystyle\int_2^3 \dfrac{x^2+2}{(x-1)(x+2)}\,dx\).

(a) \(x^2 + 2 = (x-1)(x+2) + A(x+2) + B(x-1)\). ✓
\(x = 1\): \(3 = 3A\), \(A = 1\). \(x = -2\): \(6 = -3B\), \(B = -2\). ✓
So \(1 + \dfrac{1}{x-1} - \dfrac{2}{x+2}\). ✓
(b) \(\left[x + \ln(x-1) - 2\ln(x+2)\right]_2^3 = (3 + \ln 2 - 2\ln 5) - (2 + 0 - 2\ln 4)\). ✓
\(= 1 + \ln 2 + 2\ln\dfrac{4}{5} = 1 + \ln\dfrac{32}{25}\). ✓

Analysis

[7 marks]

Water flows into a tank at a rate of \(r\) litres per minute, where

\(r = \dfrac{4t^2+3t+9}{(2t+1)(t^2+4)}\) and \(t\) is the time in minutes.

(a) Express \(r\) in partial fractions in the form \(\dfrac{A}{2t+1} + \dfrac{Bt+C}{t^2+4}\).

(b) Find the exact volume of water that flows in during the first 2 minutes, and give it to 3 significant figures.

(a) \(4t^2 + 3t + 9 = A(t^2+4) + (Bt+C)(2t+1)\). ✓
\(t = -\dfrac{1}{2}\): \(1 - \dfrac{3}{2} + 9 = \dfrac{17}{2} = \dfrac{17}{4}A\), so \(A = 2\). ✓
\(t^2\): \(4 = A + 2B\), \(B = 1\). Constants: \(9 = 4A + C\), \(C = 1\). So \(r = \dfrac{2}{2t+1} + \dfrac{t+1}{t^2+4}\). ✓
(b) \(\displaystyle\int \dfrac{2}{2t+1}\,dt = \ln(2t+1)\); \(\displaystyle\int \dfrac{t}{t^2+4}\,dt = \dfrac{1}{2}\ln(t^2+4)\); \(\displaystyle\int \dfrac{1}{t^2+4}\,dt = \dfrac{1}{2}\tan^{-1}\dfrac{t}{2}\). ✓
\(\left[\ln(2t+1) + \dfrac{1}{2}\ln(t^2+4) + \dfrac{1}{2}\tan^{-1}\dfrac{t}{2}\right]_0^2 = \ln 5 + \dfrac{1}{2}\ln 8 + \dfrac{\pi}{8} - \dfrac{1}{2}\ln 4\). ✓
\(= \ln 5 + \dfrac{1}{2}\ln 2 + \dfrac{\pi}{8} = 2.35\) litres (3 s.f.). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.