Question

Evaluate the following integral: $\int {\tan }^{{\displaystyle \frac{3}{2}}}x{\sec }^{2}xdx$

Answer 1

Hint: To find the value of a given integral, use the substitution method to simplify the given integral by assuming $t=\tan x$. Rewrite the given integral in terms of variable ‘t’. Evaluate the value of integral using the fact that $\int x^{n}dx={\displaystyle \frac{x^{n+1}}{n+1}}$. Rewrite the value of integral in terms of ‘x’.

Complete step-by-step answer:
We have to evaluate the value of the integral $\int {\tan }^{{\displaystyle \frac{3}{2}}}x{\sec }^{2}xdx$. We observe that this is an indefinite integral. An indefinite integral is a function that takes the antiderivative of another function. It represents a family of functions whose derivatives are the function given in the integral.
To find the value of the integral, we will simplify the given integral by substitution method.
Let’s assume that $t=\tan x.....\left(1\right)$. We will now differentiate the equation. Thus, we have ${\displaystyle \frac{dt}{dx}}={\sec }^{2}x$.
Cross multiplying the terms on both sides of the equality, we have $dt={\sec }^{2}xdx.....\left(2\right)$.
Substituting equation (1) and (2) in the given integral, we have $\int {\tan }^{{\displaystyle \frac{3}{2}}}x{\sec }^{2}xdx=\int t^{{\displaystyle \frac{3}{2}}}dt$.
We know that integral of a function of the form $y=x^n$ is $\int x^{n}dx={\displaystyle \frac{x^{n+1}}{n+1}}$.
Substituting $n={\displaystyle \frac{3}{2}}$ in the above formula, we have $\int x^{{\displaystyle \frac{3}{2}}}dx={\displaystyle \frac{x^{{\displaystyle \frac{5}{2}}}}{{\displaystyle \frac{5}{2}}}}$.
Thus, we have $\int {\tan }^{{\displaystyle \frac{3}{2}}}x{\sec }^{2}xdx=\int t^{{\displaystyle \frac{3}{2}}}dt={\displaystyle \frac{t^{{\displaystyle \frac{5}{2}}}}{{\displaystyle \frac{5}{2}}}}$.
Simplifying the above expression, we have $\int {\tan }^{{\displaystyle \frac{3}{2}}}x{\sec }^{2}xdx=\int t^{{\displaystyle \frac{3}{2}}}dt={\displaystyle \frac{2}{5}}{t}^{{\displaystyle \frac{5}{2}}}$.
We will again substitute $t=\tan x$ in the above equation. Thus, we have $\int {\tan }^{{\displaystyle \frac{3}{2}}}x{\sec }^{2}xdx=\int t^{{\displaystyle \frac{3}{2}}}dt={\displaystyle \frac{2}{5}}{t}^{{\displaystyle \frac{5}{2}}}={\displaystyle \frac{2}{5}}{\tan }^{{\displaystyle \frac{5}{2}}}x$.
Hence, the value of the integral $\int {\tan }^{{\displaystyle \frac{3}{2}}}x{\sec }^{2}xdx$ is ${\displaystyle \frac{2}{5}}{\tan }^{{\displaystyle \frac{5}{2}}}x$.

Note: The substitution method is used when an integral contains some function and its first derivative. It’s important to keep in mind that the first derivative of $y=\tan x$ is ${\displaystyle \frac{dy}{dx}}={\sec }^{2}x$. Otherwise, we won’t be able to solve this question.