First slide

Evaluation of definite integrals

Question

$\int_{- 4}^{- 5} e^{(x + 5)^{2}} d x + 3 \int_{1 / 3}^{2 / 3} e^{9 {(x - \frac{2}{3})}^{2}} d x$ is

Moderate

A

0
B

–2
C

1
D

2

Solution

$\begin{array}{l} I_{1} = \int_{- 4}^{- 5} e^{(x + 5)^{2}} d x = - \int_{0}^{1} e^{t^{2}} d t (t = x + 5) \\ I_{2} = \int_{1 / 3}^{2 / 3} e^{9 (x - 2 / 3)^{2}} d x = \int_{1 / 3}^{2 / 3} e^{(3 x - 2)^{2}} d x (3 x - 2 = t) \\ = \frac{1}{3} \int_{- 1}^{0} e^{t^{2}} d t = - \frac{1}{3} \int_{1}^{0} e^{t^{2}} d t \\ = \frac{1}{3} \int_{0}^{1} e^{t^{2}} d t \\ I_{1} + 3 I_{2} = 0 \end{array}$

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