How do you do definite integrals with substitution?

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Answer 1 · 2014-09-25T01:39:35

The only thing different from indefinite integral is that you will have to convert the original lower and upper limits into the new ones.

Let us evaluate the definite integral below.

$\int_{0}^{2} 2 x e^{x^{2} + 1} d x$ $int_0^2 2xe^{x^2+1} dx$

Let $u = x^{2} + 1$ $u=x^2+1$ .
$\Rightarrow \frac{d u}{d x} = 2 x \Rightarrow d u = 2 x d u$ $Rightarrow {du}/{dx}=2x Rightarrow du=2x du$

When $x = 0$ $x=0$ , $u = {(0)}^{2} + 1 = 1$ $u=(0)^2+1=1$
When $x = 2$ $x=2$ , $u = {(2)}^{2} + 1 = 5$ $u=(2)^2+1=5$

If x goes from 0 to 2, then u goes from 1 to 5.

By Substitution,

$\int_{0}^{2} 2 x e^{x^{2} + 1} d x = \int_{1}^{5} e^{u} d u = {[e^{u}]}_{1}^{u} = e^{5} - e$ $int_0^2 2xe^{x^2+1} dx=int_1^5 e^u du=[e^u]_1^5=e^5-e$