First slide

Methods of integration

Question

$Evaluate \int \frac{d x}{\sin x \cos^{3} x}$

Moderate

A

$\log |s e c x| + \frac{\tan^{2} x}{2} + c$
B

$\log |\tan x| + \frac{\tan^{2} x}{2} + c$
C

$\log |s e c x| + \frac{s e c^{2} x}{2} + c$
D

None of these

Solution

$H e r e, t h e p o w e r o f \sin x i s - 1 a n d t h a t o f \cos x i s - 3 .$
$S i n c e t h e s u m o f p o w e r s o f \sin x a n d \cos x i s - 4, w h i c h i s e v e n a n d n e g a t i v e$
$\begin{array}{l} I = \int \frac{d x}{\sin x \cos^{3} x} \\ = \int \frac{\sec^{4} x d x}{\tan x} \\ = \int \frac{(1 + \tan^{2} x) \sec^{2} x d x}{\tan x} \end{array}$
$\begin{array}{l} L e t \tan x = z . T h e n, s e c^{2} x d x = d z . T h e r e f o r e, \\ \begin{array}{l} I = \int \frac{1 + z^{2}}{z} d z = \int (\frac{1}{z} + z) d z = \log | z | + \frac{z^{2}}{2} + c \\ = \log \tan x ∣ + \frac{\tan^{2} x}{2} + c \end{array} \end{array}$

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