Integrating by Substitution

Understanding Integration by Substitution

Grasp the concept of integration by substitution, an essential technique used to simplify certain mathematical expressions before integrating.
Recognise that this method leverages the Chain Rule. In the reverse of differentiating a composite function, an integral of a function that is a derivative of another, more manageable function, can be determined.
Understand that substitution is often applied when the integrand has the pattern af(ax + b), where a and b are constants, and f is a differentiable function.

Steps of Integration by Substitution

Familiarise yourself with the general steps in integration by substitution: Identification, substitution, integration, and resubstitution.
The identification step involves spotting a function and its derivative (or a multiple thereof) within the integrand.
In the substitution step, let u be the chosen function and replace it in the integrand. Also, express dx in terms of du.
The integration step follows, where you integrate the rewritten integrand with respect to u.
In the resubstitution step, replace u with the originally chosen function to get the result in terms of x.

Practising Integration by Substitution

Aim to become adept in identifying suitable functions for substitution.
Work on many practices exercises involving integration by substitution. This will solidify your understanding and improve speed.
Pay special attention to problems with roots, fractions and other complex expressions as they often require this method.
Always remember to write + C at the end of your integration to represent the constant of integration.

Applying Integration by Substitution

Familiarise yourself with the application of this method in solving integrals in physics or engineering problems.
Understand how it is used in finding the areas under curves or in solving differential equations.
By practising, start recognising when to use substitution and when to use alternative integration methods like integration by parts.

Remember, integration by substitution might seem tricky at first. Through practicing, you’ll soon be able to quickly identify whether this method is applicable and carry out the steps accurately.