First slide

Evaluation of definite integrals

Question

The value of $\int_{\frac{- π}{2}}^{\frac{π}{2}} \frac{\cos^{2} x}{1 + 3^{x}} d x$

Moderate

A

$2 π$
B

$4 π$
C

$\frac{π}{4}$
D

$\frac{π}{2}$

Solution

$\begin{matrix} I = \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\cos^{2} x}{1 + 3^{x}} d x \\ = \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\cos^{2} x}{1 + 3^{- x}} d x (\sin c e \int_{- a}^{a} f (x) d x = \int_{- a}^{a} f (- x) d x) \\ I = \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{3^{x} \cos^{2} x}{3^{x} + 1} d x \\ 2 I = \int_{\frac{- π}{2}}^{\frac{π}{2}} \frac{\cos^{2} x (1 + 3^{x})}{3^{x} + 1} d x = \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos^{2} x d x \\ = 2 \int_{0}^{\frac{π}{2}} \cos^{2} x d x (\sin c e \cos^{2} x i s e v e n f u n c t i o n) \\ = 2 \times \frac{1}{2} \times \frac{π}{2} \\ I = \frac{π}{4} \end{matrix}$

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