Show definite integral equals specific value (trigonometric substitution)

Show a definite integral equals a specific value using a trigonometric substitution of the form x = a sin θ, x = a cos θ, or x = a tan θ.

5 questions · Standard +0.6

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Edexcel C4 Specimen Q3
8 marks Standard +0.3
3. Use the substitution \(x = \tan \theta\) to show that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x = \frac { \pi } { 8 } + \frac { 1 } { 4 }$$ (8)
OCR H240/01 2022 June Q9
7 marks Standard +0.8
9 Use the substitution \(x = 2 \sin \theta\) to show that \(\int _ { 1 } ^ { \sqrt { 3 } } \sqrt { 4 - x ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 3 } \pi\).
OCR Further Pure Core 1 2018 March Q5
6 marks Challenging +1.2
5 By using a suitable substitution, which should be stated, show that $$\int _ { \frac { 3 } { 2 } } ^ { \frac { 5 } { 2 } } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 12 x + 13 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln ( 1 + \sqrt { 2 } )$$
Edexcel C4 Q2
8 marks Standard +0.3
Use the substitution \(x = 2\tan u\) to show that $$\int_0^2 \frac{x^2}{x^2 + 4} \, dx = \frac{1}{2}(4 - \pi).$$ [8]
OCR C4 Q6
8 marks Standard +0.3
Use the substitution \(x = 2 \tan u\) to show that $$\int_0^2 \frac{x^2}{x^2 + 4} \, dx = \frac{1}{2}(4 - \pi).$$ [8]