Maths and Statistics

Introductory Maths

Substitution by integration

As a first year undergraduate student you may have come across substitution by integration (a part of calculus).

In the video (below), Dr Janet Bonar (Course Leader in the School of Maritime Science and Engineering) demonstrates a step by step approach to solving the following problem:

\( F( x )= \int_{} x^{2} \) cos(\( \chi \)3 + 1)  d\( \chi \), using the substitution u= \( \chi \)3 + 1


Try and solve the problem yourself by following the steps below, before watching the video.

Step 1 When two functions are multiplied and you want to integrate, you have two techniques you can use: integration by substitution, or integration by parts.  In general, try integration by substitution first to see if it will work, as it’s a bit tidier.

Step 2 Guess a likely substitution.   If there are brackets, this is often the bit inside the brackets.  Take the derivative of the substitution.  If the derivative term is in the original equation, then substitution is the right technique.  In particular, the variable should be in your derivative, and to the same power, as in the other part of your original equation.

Step 3  To do the substitution, solve your derivative term \( \frac{du}{dx} \) for d\( \chi \)

Step 4 Substitute your u term and your d\( \chi \) term into the original equation and cancel where you can.  

Have you changed all instances of your original variable to your new variable?
Now you can see why it's called 'substitution'!

Step 5 Integrate your new, simple equation.  It should be much easier!

Step 6 Write the solution, don’t forget to include the constant of integration.

Step 7 Substitute the original variable back in—and you’re done!


Now watch how Dr Janet Bonar works through the steps in this video: