2.1: Substitution
- Page ID
- 521
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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:
- Find the function derivative pair (\(f\) and \(f'\)).
- Let \(u = f(x)\).
- Find \(du/dx\) and adjust for constants.
- Substitute.
- Integrate.
- Resubstitute.
We will try many more examples including those such as
\[ \int x\, \sin(x^2)\, dx, \]
\[ \int x\, \sqrt{x - 2}\, dx. \]
Contributors and Attributions
- Larry Green (Lake Tahoe Community College)
Integrated by Justin Marshall.