Indefinite integral with Weierstrass or trigonometric substitution

Find an indefinite integral using a trigonometric substitution such as t = tan(x/2) (Weierstrass) or similar, where the integrand involves trigonometric functions, returning an answer in terms of x.

2 questions · Challenging +1.2

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OCR FP2 2011 January Q1

5 marks Challenging +1.2

1 Use the substitution \(t = \tan \frac { 1 } { 2 } x\) to find \(\int \frac { 1 } { 1 + \sin x + \cos x } \mathrm {~d} x\).

Edexcel FP1 2019 June Q5

8 marks Challenging +1.2

5. $$I = \int \frac { 1 } { 4 \cos x - 3 \sin x } \mathrm {~d} x \quad 0 < x < \frac { \pi } { 4 }$$ Use the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) to show that $$I = \frac { 1 } { 5 } \ln \left( \frac { 2 + \tan \left( \frac { x } { 2 } \right) } { 1 - 2 \tan \left( \frac { x } { 2 } \right) } \right) + k$$ where \(k\) is an arbitrary constant.