2.1: Substitution

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Oct 27, 2024
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- 2: Techniques of Integration
- 2.2: Trigonometric Substitution

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Recall that the chain rule states that

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

Integrating both sides we get:

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

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

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:

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

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

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

Contributors and Attributions

Larry Green (Lake Tahoe Community College)
Integrated by Justin Marshall.

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Solution

Example 1

Solution

Contributors and Attributions

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