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2.1: Substitution

[ "article:topic", "integration by substitution", "authorname:green" ]

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Recall that the chain rule states that

$(f(g(x)))' = f'(g(x))g'(x).$

Integrating both sides we get:

$\int[f(g(x)]'dx = \int[f'(g(x)g'(x)dx]$

or

$\int f'\left( g(x) \right) \, g' (x) \, dx = f\left(g(x)\right) + C$

Example 1

Calculate

$\int \dfrac{2x}{x^2+1}\, dx = \int 2x\left( x^2+1\right)^{-2} \, dx.$

Solution

Let

$u = x^2 +1$

then

$\dfrac{du}{dx} = 2x$

and

$du = 2x \,dx.$

We substitute:

$\int u^{-2} du = -u^{-1} + C = (x^2 +1)^{-1} + C.$

Steps:

1. Find the function derivative pair ($$f$$ and $$f'$$).
2. Let $$u = f(x)$$.
3. Find $$du/dx$$ and adjust for constants.
4. Substitute.
5. Integrate.
6. Resubstitute.

We will try many more examples including those such as

$\int x\, \sin(x^2)\, dx,$

$\int x\, \sqrt{x - 2}\, dx.$