Question

Evaluate the value of $\int {x{{\sec }^2}xdx} $ .

Answer 1

Hint: Firstly, let $\int {x{{\sec }^2}xdx} = I$ .
Then, find the value of integral using the formula $\int {f\left( x \right)g\left( x \right)dx} = f\left( x \right)\int {g\left( x \right)dx} - \int {\left[ {\dfrac{d}{{dx}}f\left( x \right) \cdot \int {g\left( x \right)dx} } \right]dx} $ , here $f\left( x \right) = x$ and $g\left( x \right) = {\sec ^2}x$ .
Thus, find the value of I and hence we found the required answer.

Complete step-by-step answer:
Here, we are asked to find the value of $\int {x{{\sec }^2}xdx} $ .
Let $\int {x{{\sec }^2}xdx} = I$ .
 $\Rightarrow I = \int {x{{\sec }^2}xdx} $
Since, we know that the integral of the product of two functions \[f\left( x \right)\] and \[g\left( x \right)\] is $\int {f\left( x \right)g\left( x \right)dx} = f\left( x \right)\int {g\left( x \right)dx} - \int {\left[ {\dfrac{d}{{dx}}f\left( x \right) \cdot \int {g\left( x \right)dx} } \right]dx} $ .
So, we will find the integral of I, by using the above formula, where $f\left( x \right) = x$ and $g\left( x \right) = {\sec ^2}x$ .
 $\Rightarrow I = x \cdot \int {{{\sec }^2}xdx} - \int {\left[ {\dfrac{{dx}}{{dx}} \cdot \int {{{\sec }^2}xdx} } \right]dx} $
 $\Rightarrow I = x\tan x - \int {1 \cdot \tan xdx} $
 $\Rightarrow I = x\tan x - \int {\tan xdx} $
 $\Rightarrow I = x\tan x - \log \left| {\sec x} \right| + C$
Thus, $\int {x{{\sec }^2}xdx} = x\tan x - \log \left| {\sec x} \right| + C$ .

Note: The formula for integration by parts is given by $\int {f\left( x \right)g\left( x \right)dx} = f\left( x \right)\int {g\left( x \right)dx} - \int {\left[ {\dfrac{d}{{dx}}f\left( x \right) \cdot \int {g\left( x \right)dx} } \right]dx} $ .
In the formula for integration by parts, \[f\left( x \right)\] and \[g\left( x \right)\] are chosen by the method of ILATE.
ILATE is known as Inverse trigonometric functions, Logarithm functions, Algebraic functions, Trigonometric functions and Exponential functions.
The function \[f\left( x \right)\] must be a function lying before the function \[g\left( x \right)\] in the ILATE rule.
In the given question, we chose $f\left( x \right) = x$ and $g\left( x \right) = {\sec ^2}x$ , because x is an algebraic function and ${\sec ^2}x$ is an trigonometric function and according to ILATE rule algebraic functions lie before trigonometric functions.