Definite Integration

Integration is sometimes called antidifferentiation, as it is the opposite process of differentiation. When we find an indefinite integral, we find a function with an arbitrary constant, C. Whereas, when we find a definite integral, we find a numerical value. Definite integration has many applications. For the examination, we use it to find displacement in questions about kinematics, probabilities with continuous random variables and it is linked to work on finding areas and volumes with integration.


Key Concepts

On this page, you should learn about

  • definite integrals
    • \(\int_{a}^{b} f'(x)\: dx=\: f(b)-f(a)\)

Essentials

The following videos will help you understand all the concepts from this page

What and Why

In the following video, we are going to look at what definite integration is (comparing to indefinite integration) and what we use it for. We'll look at some of the many applications of definite integration.

How to do it

The process of finding definite integrals is relatively straightforward, but it is important to carefully use the correct notation. In the following video, we will look at two examples:  \(\int_{\frac{\pi}{2}}^{\pi} cosx\: dx\)   and   \(\int_{1}^{e} \frac{1}{x}\: dx\)

Summary

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Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

Exam-style Questions

Question 1

Given that \(\int _{ 4 }^{ 8 }{ \frac { 1 }{ 2x-4 } dx= } ln\sqrt { a } \) , find the value of a

Hint

Full Solution

Question 2

Consider a function f(x) such that \(\int _{ 0 }^{ 4 }{ f(x)dx } \) = 6

Find

a) \(\int _{ 0 }^{ 4 }{3 f(x)dx } \)

b) \(\int _{ 0 }^{ 4 }{[ f(x)+3]dx } \)

c) \(\int _{ -3 }^{ 1 }{\frac{1}{3} f(x+3)dx } \)

d) \(\int _{ 0 }^{ 4 }{ [f(x)+x]dx } \)

Hint

Full Solution

Question 3

Given that \(\int _{ 2 }^{ 5 }{ ln(sinx)dx=A } \)

show that \(\int _{ 2 }^{ 5 }{ ln({ e }^{ x }sinx)dx=A } +\frac { 21 }{ 2 } \)

Hint

Full Solution

MY PROGRESS

How much of Definite Integration have you understood?