2.1: Substitution

Recall that the chain rule states that

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

Integrating both sides we get:

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

or

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

Example 1

Calculate

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

Solution

Let

$$u = x^2 +1 \nonumber$$

then

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

and

$$du = 2x \,dx. \nonumber$$

We substitute:

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

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

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

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

Contributors and Attributions

• Larry Green (Lake Tahoe Community College)

• Integrated by Justin Marshall.