First slide

Evaluation of definite integrals

Question

If $f (x) = \int_{x^{2}}^{x^{2} + 4} e^{- t^{2}} d t,$ then the function $f (x)$ increases in

Moderate

A

$(- \infty, 0)$
B

$(0, \infty)$
C

$(- 1, 2)$
D

$(- 2, \infty)$

Solution

$\begin{array}{l} f^{'} (x) = 2 x e^{- {(x^{2} + 4)}^{2}} - 2 x e^{- x^{4}} \\ = 2 x e^{- {(x^{2} + 4)}^{2}} [1 - e^{16 + 8 x^{2}}] \end{array}$
Since $1 - e^{16 + 8 x^{2}} < \bar{0}$ for all $x$ so $f$ increases for $x < 0 .$

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