Integration by Substitution SL

  2. Calculus
  3. Integration by Substitution SL
    4. 30'

Integration by substitution or U-substitution is a method that will help you integrate many different functions. By changing the variable of the integrand, we can make an apparently difficult problem into a much simpler one. The challenge is recognising when it is a useful method and what the substitution is (it is very rare that this will be suggested to you). On this page, we'll concentrate on doing that. In order to master the techniques, it is vital that you get plenty of practice!

Key Concepts

On this page, you should learn about

  • integration by substitution
 

Essentials

In the videos, we will look at the method and how to set it up, but will also concentrate on how we recognise that it is an integration by substitution question.

Example 1

\(\int { \frac { cosx }{ { \left( sinx-1 \right) }^{ 2 } } dx } \)

Example 2

\(\int { 2{ e }^{ x }cos({ e }^{ x })dx } \)

 

Definite Integral

When we are required to evaluate definite integrals then care is required dealing with the limits of integration. It is important to change the limits of integration according to the substitution that you use. The video below shows you how to do that

 \(\int _{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 2 } }{ { cos }^{ 3 }xsinxdx } \)

 

 

Difficult Example

\(\int { { sin }^{ 3 }xdx } \)

 

Summary

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Test Yourself

The challenge when you start doing Integration by substitution questions is choosing what the correct substitution is. Remember, you are trying to find a function f(x) within the integrand such that the differential of this, f'(x) is also present in the integrand

\(\int { g'(f(x)) \ \cdot f'(x)\quad \ dx } \)


START QUIZ!  

Here's the chance to practise the techniques of integration by substitution


START QUIZ!

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Exam-style Questions

Question 1

a) Find \(\int { \frac { { e }^{ x } }{ 1+{ e }^{ x } } dx } \)

b) Evaluate \(\int _{ \frac { { \pi }^{ 2 } }{ 4 } }^{ \pi ^{ 2 } }{ \frac { cos\sqrt { x } }{ \sqrt { x } } } dx\)

Hint

Full Solution

 

Video Solution

 

Question 2

The following diagram shows the graph of f(x) = \(\frac { 4x }{ \sqrt { { x }^{ 2 }+1 } }\)

Let R be the region bounded by f, the x-axis, x = 1 and x = 2

Find R

Hint

Full Solution

 

Video Solution

 

Question 3

a) Using the fact that \(tanx = \frac{sinx}{cosx} \), show that \(\frac{d}{dx}(tanx) = \frac{1}{cos^2x}\)

b) Hence, find \(\int { \frac { \sqrt { tanx } }{ cos^{ 2 }x } } dx\)

Hint

Full Solution

 

Video Solution

 
 
MY PROGRESS

How much of Integration by Substitution SL have you understood?