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Chapter 5: Q 65. (page 429)

Solve the integral: $鈭� \frac{x}{\cos^{2} x} d x$

Short Answer

Expert verified

The required answer is $x \tan x + \ln (\cos x) + c$ .

Step by step solution

01

Step 1. Given information.

We have given integral is $鈭� \frac{x}{\cos^{2} x} d x$ .

02

Step 2. Solve the integration by parts.

We have,

$u = x d u = d x$

and

$d v = \frac{d x}{\cos^{2} x} v = 鈭� \frac{1}{\cos^{2} x} d x v = \tan x$

The formula of integration by parts is $鈭� u d v = u v - 鈭� v d u$ .

$鈭� \frac{x}{\cos^{2} x} d x = x \tan x - 鈭� \tan x . 1 d x = x \tan x - 鈭� \tan x d x = x \tan x - 鈭� \frac{\sin x}{\cos x} = x \tan x - 鈭� (- \frac{1}{u}) = x \tan x + \ln (u) + c = x \tan x + \ln (\cos x) + c$

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