The value of $\int \frac{\sec x \cdot an x}{9-16 an ^2 x} \mathrm{~d} x$ is equal to

Question

The value of $\int \frac{\sec x \cdot 	an x}{9-16 	an ^2 x} \mathrm{~d} x$ is equal to

Aicharya · Accepted Answer

Let I $\begin{aligned} \mathrm{I} & =\int \frac{\sec x \tan x}{9-16 \tan ^2 x} \mathrm{~d} x \\ & =\int \frac{\sec x \tan x}{9-16\left(\sec ^2 x-1\right)} \mathrm{d} x \\ & =\int \frac{\sec x \tan x}{25-16 \sec ^2 x} \mathrm{~d} x \end{aligned}$
Put $\sec x=\mathrm{t} \Rightarrow \sec x \tan x \mathrm{~d} x=\mathrm{dt}$ $\begin{aligned} \therefore \quad \mathrm{I} & =\int \frac{\mathrm{dt}}{5^2-(4 \mathrm{t})^2} \\ & =\frac{1}{2(5)} \cdot \frac{1}{4} \log \left|\frac{5+4 \mathrm{t}}{5-4 t}\right| \\ & =\frac{1}{40} \log \left|\frac{5+4 \sec x}{5-4 \sec x}\right|+\mathrm{c} \end{aligned}$

The value of $\int \frac{\sec x \cdot \tan x}{9-16 \tan ^2 x} \mathrm{~d} x$ is equal to

Solution