First slide

Introduction to definite integration

Question

The value of $\int_{\frac{π}{6}}^{\frac{π}{4}} \frac{d x}{1 + e^{\sin x}}$

Moderate

A

$π$
B

$\frac{3 π}{2}$
C

$\frac{π}{4}$
D

$\frac{π}{2}$

Solution

$\begin{matrix} I = \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1}{1 + e^{\sin x}} d x \\ = \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1}{1 + e^{- \sin x}} d x (∵ \int_{- a}^{a} f (x) d x = \int_{- a}^{a} f (- x) d x) \\ 2 I = \int_{- \frac{π}{2}}^{\frac{π}{2}} d x \\ = π \end{matrix}$
Hence, $I = \frac{π}{2}$

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