First slide

Evaluation of definite integrals

Question

$\int_{0}^{π} e^{\sin^{2} x} \cos^{3} xdx$ is equal to

Moderate

A

-1
B

0
C

1
D

$π$

Solution

$\begin{array}{l} Let I = \int_{0}^{π} e^{\sin^{2} x} \cos^{3} xdx - - - - - (i) \\ \Rightarrow I = \int_{0}^{π} e^{\sin^{2} (π - x)} \cos^{3} (π - x) dx \\ \Rightarrow I = - \int_{0}^{π} e^{\sin^{2} x} \cos^{3} xdx - - - - - (ii) \end{array}$
On adding Eqs. (i) and (ii), we get
$2 I = 0 \Rightarrow I = 0$

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