Question

Evaluate the integral: \(\int {\tan x{{\sec }^3}xdx} \).

Short answer

To evaluate the integral:\(\int {\tan x{{\sec }^3}xdx} \), first we will split\({\sec ^3}x = \sec x{\sec ^2}x\)and then usethe substitution method.

\(\int {\tan x{{\sec }^3}} xdx = \frac{1}{3}{\sec ^3}x + c\)

Step-by-step solution

Splitting the third power of the secant function

Let I =\(\int {\tan x{{\sec }^3}xdx} \)

Write\({\sec ^3}x = \sec x{\sec ^2}x\)

\(I = \int {\tan x\sec x{{\sec }^2}} xdx\)

Substitution Method

Now we will use the substitution method

Put

\(\begin{aligned}{l}\sec x = u\\ = \sec x\tan xdx = du\\I = \int {{u^2}} du = \frac{1}{3}{u^3} + c\end{aligned}\)

Re-Substitution of the value of u

Now Re-Substitute the value of\(u = \sec x\)

\(I = \frac{1}{3}{\sec ^3}x + c\)

Hence,the value of integral:

\(\int {\tan x{{\sec }^3}} xdx = \frac{1}{3}{\sec ^3}x + c\)