Basic Integration

What is integration
ntegration is the reverse process of differentiation. It helps find the antiderivative (original function) from its derivative.

The following diagrams show some examples of Integration Rules: Power Rule, Exponential Rule, Constant Multiple, Absolute Value, Sums and Difference. Scroll down the page for more examples and solutions on how to integrate using some rules of integrals.

How to Solve Basic Integration Problems:

1. Identify the integrand: This is the function you are trying to integrate.
2. Apply the appropriate rule(s): Use the basic rules mentioned above to find the antiderivative of each term in the integrand.
3. Include the constant of integration: Always add (+ C) to the end of your result for indefinite integrals.
4. Simplify (if possible): Combine any like terms or simplify the expression.
5. You can always check your answer by differentiating it. The derivative of your antiderivative should be the original integrand.

Example of Basic Integration:
\(\int (4x^3 - 6x^2 + 2x + 5) , dx\)
\( = \int 4x^3 , dx - \int 6x^2 , dx + \int 2x , dx + \int 5 , dx\) (Sum/Difference Rule)
\( = 4 \int x^3 , dx - 6 \int x^2 , dx + 2 \int x , dx + 5 \int 1 , dx\) (Constant Multiple Rule)
\( = 4 \left( \frac{x^4}{4} \right) - 6 \left( \frac{x^3}{3} \right) + 2 \left( \frac{x^2}{2} \right) + 5(x) + C\) (Power Rule, Constant Rule)
\( = x^4 - 2x^3 + x^2 + 5x + C\)

Basic Integration
This video will show:
How to find antiderivatives, or indefinite integrals, using basic integration rules.
How to integrate a polynomial with the power rule?
How to integrate negative powers of x, fractional powers of x, and radicals/roots
How to integrate x^-1, or 1/x, with the log rule
How to use algebra to rewrite before integrating, separate the numerator of a fraction, or expand a product in order to use the power rule.
Basic Trig and Exponential examples that use rules from the Table of Integrals, as well as trig identities.

• Show Video Lesson

Edexcel C1 Core Maths June 2007 Question 9
The curve C with equation y = f(x) passes through the point (5, 65). Given that f’(x) = 6x² - 10x - 12,
(a) use integration to find f(x).
(b) Hence show that f(x) = x(2x + 3)(x - 4).
(c) In the space provided on page 17, sketch C, showing the coordinates of the points where C crosses the x-axis.

• Show Video Lesson

A-Level Maths Edexcel C1 June 2009 Q3b
Worked solution to the above core 1 question on integration.
Integrate
y = 2x³ + 3/x², x ≠ 0

• Show Video Lesson

A-Level : Edexcel : C1 June 2008 Q1
Short question on basic integration.
Integrate 2 + 5x²

• Show Video Lesson

A-Level Edexcel C2 June 2007 Q1
Worked solution to this question on integration.

• Show Video Lesson

Try out our new and fun Fraction Concoction Game.

Add and subtract fractions to make exciting fraction concoctions following a recipe. There are four levels of difficulty: Easy, medium, hard and insane. Practice the basics of fraction addition and subtraction or challenge yourself with the insane level.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.