Show integral transforms via substitution then evaluate (algebraic/exponential)

A two-part question where (i) a substitution transforms the integral to a new algebraic or exponential form, and (ii) the transformed integral is evaluated to a specific exact value, without requiring trigonometric substitutions or the Weierstrass t-substitution.

4 questions · Standard +0.3

Sort by: Default | Easiest first | Hardest first

CAIE P3 2015 June Q6

8 marks Standard +0.3

6 Let $I = \int _ { 0 } ^ { 1 } \frac { \sqrt { } x } { 2 - \sqrt { } x } \mathrm {~d} x$.

Using the substitution $u = 2 - \sqrt { } x$, show that $I = \int _ { 1 } ^ { 2 } \frac { 2 ( 2 - u ) ^ { 2 } } { u } \mathrm {~d} u$.
Hence show that $I = 8 \ln 2 - 5$.

CAIE P3 2016 June Q7

8 marks Standard +0.3

7 Let $I = \int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { \left( 1 + x ^ { 2 } \right) ^ { 3 } } \mathrm {~d} x$.

Using the substitution $u = 1 + x ^ { 2 }$, show that $I = \int _ { 1 } ^ { 2 } \frac { ( u - 1 ) ^ { 2 } } { 2 u ^ { 3 } } \mathrm {~d} u$.
Hence find the exact value of $I$.

CAIE P3 2023 June Q7

8 marks Standard +0.3

Use the substitution $u = \cos x$ to show that $$\int _ { 0 } ^ { \pi } \sin 2 x \mathrm { e } ^ { 2 \cos x } \mathrm {~d} x = \int _ { - 1 } ^ { 1 } 2 u \mathrm { e } ^ { 2 u } \mathrm {~d} u$$
Hence find the exact value of $\int _ { 0 } ^ { \pi } \sin 2 x \mathrm { e } ^ { 2 \cos x } \mathrm {~d} x$.

OCR C4 2006 June Q6

8 marks Standard +0.3

Show that the substitution $u = e^x + 1$ transforms $\int \frac{e^{2x}}{e^x + 1} dx$ to $\int \frac{u - 1}{u} du$. [3]
Hence show that $\int_0^1 \frac{e^{2x}}{e^x + 1} dx = e - 1 - \ln\left(\frac{e + 1}{2}\right)$. [5]