First slide

Introduction to definite integration

Question

$Evaluate \int_{0}^{4 π} \frac{d x}{\cos^{2} x (2 + \tan^{2} x)}$

Moderate

A

$\sqrt{2} π$
B

2 $\sqrt{2} π$
C

3 $\sqrt{2} π$
D

4 $\sqrt{2} π$

Solution

$\begin{matrix} \int_{0}^{4 π} \frac{\sec^{2} x}{(2 + \tan^{2} x)} d x = 2 \int_{0}^{2 π} \frac{\sec^{2} x d x}{2 + \tan^{2} x} u \sin g f o r m u l a \int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (x) d x i f f (2 a - x) = f (x) \\ = 4 \int_{0}^{π} \frac{\sec^{2} x}{2 + \tan^{2} x} d x \\ = 8 \int_{0}^{\frac{π}{2}} \frac{\sec^{2} x d x}{2 + \tan^{2} x} \\ = 8 \int_{0}^{π / 2} \frac{d (\tan x)}{2 + \tan^{2} x} \\ = \frac{8}{\sqrt{2}} \tan^{- 1} {(\frac{\tan x}{\sqrt{2}})}_{0}^{π / 2} \\ = \frac{8}{\sqrt{2}} (\frac{π}{2} - 0) = 2 \sqrt{2} π \end{matrix}$

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