Parametric or substitution with partial fractions

Apply a substitution or work with parametric equations, then use partial fractions to evaluate an integral or solve a differential equation.

2 questions · Standard +0.8

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OCR FP2 2007 January Q7
9 marks Challenging +1.2
7
  1. Express \(\frac { 1 - t ^ { 2 } } { t ^ { 2 } \left( 1 + t ^ { 2 } \right) }\) in partial fractions.
  2. Use the substitution \(t = \tan \frac { 1 } { 2 } x\) to show that $$\int _ { \frac { 1 } { 3 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { \cos x } { 1 - \cos x } \mathrm {~d} x = \sqrt { 3 } - 1 - \frac { 1 } { 6 } \pi$$
OCR C4 Q9
14 marks Standard +0.3
  1. Show that the substitution \(u = \sin x\) transforms the integral $$\int \frac{6}{\cos x(2 - \sin x)} dx$$ into the integral $$\int \frac{6}{(1-u^2)(2-u)} du.$$ [4]
  2. Express \(\frac{6}{(1-u^2)(2-u)}\) in partial fractions. [4]
  3. Hence, evaluate $$\int_0^{\pi/6} \frac{6}{\cos x(2 - \sin x)} dx,$$ giving your answer in the form \(a \ln 2 + b \ln 3\), where \(a\) and \(b\) are integers. [6]