First slide

Introduction to integration

Question

$\int \frac{\sec^{2} x - 7}{\sin^{7} x} d x =$

Moderate

A

$\frac{\tan x}{(\sin x)^{7}} + C$
B

$\frac{\cos x}{\sin^{7} x} + C$
C

$\frac{\sin x}{\cos^{7} x} + C$
D

$None of these$

Solution

$\begin{array}{l} I = \int \frac{\sec^{2} x - 7}{\sin^{7} x} dx \\ = \int \frac{\sec^{2} x}{\sin^{7} x} dx - 7 \int \frac{dx}{\sin^{7} x} \\ = \frac{\tan x}{\sin^{7} x} + 7 \int \frac{\tan xcos x}{\sin^{8} x} dx - 7 \int \frac{dx}{\sin^{7} x} \\ = \frac{\tan x}{\sin^{7} x} + C \end{array}$

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