The Change of Variables Theorem

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.

When we were learning differential calculus, we learned that we could differentiate more complicated functions by using the chain rule to break a function up into two smaller functions which recognised and could then differentiate directly. The Change of Variables Theorem is a technique that is a bit like 'the chain rule for integration'. It shows us how we can change the function inside the integration by changing the variables we are integrating over.

\int g(u(x)) \cdot u'(x) dx = \int g(y) dy \quad \text{where } y = u(x)

While this technique is not as straightforward as the chain rule for differentiation, we will see that we can use this theorem to transform integrals either by going from left to right, or by going from right to left.

The Change of Variables Theorem and is a fundamental result in calculus, which applies to a wider range of integrals than those we have explored so far. The theorem also specifies certain conditions that $u(x)$ must meet for the theorem to hold, and we will explore what happens if it does not later.

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