Integration By $u$-Substitution

Most of the integration we have done in the past has been based on 'just knowing' what the pattern is to convert the function inside an integral into the answer. For example, when we see $\int x^6 dx$ we remember to increase the power by $1$, and then divide the expression by this new number to obtain $\int x^6 dx = \frac{x^{6+1}}{6+1} = \frac{x^7}7$

Sometimes, when we have a complicated function inside an integral, we don't know the pattern (or it doesn't exist) so we need to rely on more advanced techniques.

Integration by substitution is a technique that is a bit like 'the chain rule for integration'. If we can re-write the expression inside the integral using some 'intermediate' variable $u$, where $u$ is itself a function of $x$, then we can look for a pattern that resembles the derivate we would obtain by using the chain rule.

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