Show definite integral equals specific value (algebraic/exponential substitution)

Show a definite integral equals a specific value using a substitution where the integrand involves algebraic or exponential functions (e.g. u = linear, polynomial, exponential, or square root function), without requiring trigonometric identities.

CAIE P3 2011 November Q10

10 marks Challenging +1.2

Use the substitution $u = \tan x$ to show that, for $n \neq - 1$, $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { n + 2 } x + \tan ^ { n } x \right) \mathrm { d } x = \frac { 1 } { n + 1 }$$
Hence find the exact value of
1. $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \sec ^ { 4 } x - \sec ^ { 2 } x \right) \mathrm { d } x$,
2. $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 9 } x + 5 \tan ^ { 7 } x + 5 \tan ^ { 5 } x + \tan ^ { 3 } x \right) \mathrm { d } x$.

Edexcel P3 2022 January Q3

6 marks Moderate -0.3

3. (i) Find, in simplest form, $$\int ( 2 x - 5 ) ^ { 7 } \mathrm {~d} x$$ (ii) Show, by algebraic integration, that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { 4 \sin x } { 1 + 2 \cos x } \mathrm {~d} x = \ln a$$ where $a$ is a rational constant to be found.

Edexcel C34 Specimen Q3

6 marks Standard +0.3

Using the substitution $u = \cos x + 1$, or otherwise, show that

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { ( \cos x + 1 ) } \sin x \mathrm {~d} x = \mathrm { e } ( \mathrm { e } - 1 )$$

Show that

$$\int _ { 2 } ^ { 4 } x \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = 8 \sqrt { 3 }$$

OCR C3 2012 June Q4

8 marks Moderate -0.3

Show that $\int _ { 0 } ^ { 4 } \frac { 18 } { \sqrt { 6 x + 1 } } \mathrm {~d} x = 24$.
Find $\int _ { 0 } ^ { 1 } \left( \mathrm { e } ^ { x } + 2 \right) ^ { 2 } \mathrm {~d} x$, giving your answer in terms of e .

OCR MEI C3 2013 June Q6

5 marks Standard +0.3

6 Using a suitable substitution or otherwise, show that $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \sin 2 x } { 3 + \cos 2 x } \mathrm {~d} x = \frac { 1 } { 2 } \ln 2$.

OCR C4 2012 June Q6

7 marks Standard +0.3

6 Use the substitution $u = 1 + \sqrt { x }$ to show that $$\int _ { 4 } ^ { 9 } \frac { 1 } { 1 + \sqrt { x } } \mathrm {~d} x = 2 + 2 \ln \frac { 3 } { 4 }$$

AQA C3 2006 January Q3

10 marks Standard +0.3

1. Given that $\mathrm { f } ( x ) = x ^ { 4 } + 2 x$, find $\mathrm { f } ^ { \prime } ( x )$.
2. Hence, or otherwise, find $\int \frac { 2 x ^ { 3 } + 1 } { x ^ { 4 } + 2 x } \mathrm {~d} x$.
1. Use the substitution $u = 2 x + 1$ to show that $$\int x \sqrt { 2 x + 1 } \mathrm {~d} x = \frac { 1 } { 4 } \int \left( u ^ { \frac { 3 } { 2 } } - u ^ { \frac { 1 } { 2 } } \right) \mathrm { d } u$$
2. Hence show that $\int _ { 0 } ^ { 4 } x \sqrt { 2 x + 1 } \mathrm {~d} x = 19.9$ correct to three significant figures.