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Added a section on the reverse chain rule
:<big><math>\frac{d}{dx}ln|x^2+3|=\left (\frac{1}{x^2+3} \right)(2x)=\frac{2x}{x^2+3}</math></big>
:<big><math>\frac{d}{dx}ln|x^2+3|=\left (\frac{1}{x^2+3} \right)(2x)=\frac{2x}{x^2+3}</math></big>
===Reverse Chain Rule===
The reverse chain rule is a trick for integration by substitution without having to do all the steps involved with integration by substitution. It uses a reversed form of the [[chain rule]] in [[differentiation]] to integrate functions of the form f'(x)g'(f(x)), where f and g are two [[function]]s and f'(x) is the derivative of f(x). From the chain rule, which states:
:<math>
\frac{d}{dx} \left( g(f(x)) \right) = f'(x) g'(f(x))
</math>
Integrating both sides produces the reverse chain rule:
:<math>
\int f'(x) g'(f(x)) \, dx = g(f(x))
</math>
This means that if a function can be recognised in the form f'(x)g'(f(x)), then it can be integrated almost immeadiately. As an example, consider the function:
:<math>
y(x) = 2x \cos{\left( x^2 \right)}
</math>
Comparing to the above, f(x)=x<sup>2</sup>, f'(x)=2x and g'(x) is represented by a [[cosine]]. All wee need to do is know the integral of a cosine is [[sine]] so g(x) is sine. Therefore:
:<math>
\int 2x \cos{\left( x^2 \right)} \, dx = \sin{\left( x^2 \right)}
</math>
If the first part of the function is f'(x) multiplied by a constant, then this constant can be taken out of the integral so that it the integral is of the form f'(x)g'(f(x)). This method can also be applied to the fraction example above.
==Trigonometric Substitution==
==Trigonometric Substitution==