QUESTION

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

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 $\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}$. Rewrite the value of integral in terms of ‘x’.

We have to evaluate the value of the integral $\int{{{\tan }^{\dfrac{3}{2}}}x{{\sec }^{2}}xdx}$. 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.
Let’s assume that $t=\tan x.....\left( 1 \right)$. We will now differentiate the equation. Thus, we have $\dfrac{dt}{dx}={{\sec }^{2}}x$.
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 $\int{{{\tan }^{\dfrac{3}{2}}}x{{\sec }^{2}}xdx}=\int{{{t}^{\dfrac{3}{2}}}dt}$.
We know that integral of a function of the form $y={{x}^{n}}$ is $\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}$.
Substituting $n=\dfrac{3}{2}$ in the above formula, we have $\int{{{x}^{\dfrac{3}{2}}}dx}=\dfrac{{{x}^{\dfrac{5}{2}}}}{\dfrac{5}{2}}$.
Thus, we have $\int{{{\tan }^{\dfrac{3}{2}}}x{{\sec }^{2}}xdx}=\int{{{t}^{\dfrac{3}{2}}}dt}=\dfrac{{{t}^{\dfrac{5}{2}}}}{\dfrac{5}{2}}$.
Simplifying the above expression, we have $\int{{{\tan }^{\dfrac{3}{2}}}x{{\sec }^{2}}xdx}=\int{{{t}^{\dfrac{3}{2}}}dt}=\dfrac{2}{5}{{t}^{\dfrac{5}{2}}}$.
We will again substitute $t=\tan x$ in the above equation. Thus, we have $\int{{{\tan }^{\dfrac{3}{2}}}x{{\sec }^{2}}xdx}=\int{{{t}^{\dfrac{3}{2}}}dt}=\dfrac{2}{5}{{t}^{\dfrac{5}{2}}}=\dfrac{2}{5}{{\tan }^{\dfrac{5}{2}}}x$.
Hence, the value of the integral $\int{{{\tan }^{\dfrac{3}{2}}}x{{\sec }^{2}}xdx}$ is $\dfrac{2}{5}{{\tan }^{\dfrac{5}{2}}}x$.
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 $\dfrac{dy}{dx}={{\sec }^{2}}x$. Otherwise, we won’t be able to solve this question.