Integration by Substitution HL

  2. Calculus
  3. Integration by Substitution HL
    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

The first videos are fairly simple examples which use Integration by substitution. 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.

Simple Example 1

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

Simple Example 2

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

 

Simple Example 3

\(\int { \frac { arctanx }{ x^{ 2 }+1 } } dx\)

 

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 1 

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

 

Difficult Example 2

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

 

Difficult Example 3

It is really not obvious at first glance how Integration by substitution helps with the following question. The video concentrates on how you might go through the decision making to decide to use this method. Interestingly, this question can also be solved using Integration by parts. A challenge for you to try for yourself!

\(\int { x\sqrt { x+1 } dx } \)

 

Difficult Example 4

In the following example, the substitution has been suggested for us. It would appear to make it an easier type of question. However, it is not! Very careful manipulation is required to get to the correct result.

Find \(\int { \frac { x^{ 2 } }{ \sqrt { 36-x^ 2 } } dx } \) using the substitution \( x=sin\theta\)

 

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

 

Question 4

Find \(\int { \frac { arcsinx+x }{ \sqrt { 1-x^{ 2 } } } } dx\)

Hint

Full Solution

 

Video Solution

 

Question 5

Find \(\int { sin^{ 5 }x\ dx } \)

Hint

Full Solution

 

Video Solution

 

Question 6

Find \(\int { \sqrt { 16-9x^{ 2 } } dx } \) using the substitution 3x = 4 sin\(\theta\)

Hint

Full Solution

 

Video Solution

 
 
MY PROGRESS

How much of Integration by Substitution HL have you understood?