Question

How do you integrate $x{\sec ^2}x$ ?

Answer 1

For integrating $x{\sec ^2}x$ , this involve two functions, for this we have to use integration by parts that is , $\int {u \cdot vdx = u\int {vdx - \int {\left( {\dfrac{{du}}{{dx}}\int {vdx} } \right)} } } dx$ . Here the criteria for selecting first function $u$ is according to the $ILATE$ rule , where $I$ denotes inverse Trigonometric Function, $L$ denotes Logarithm Function, $A$ denotes Algebraic Function, $T$ denotes Trigonometric Function,
$E$ denotes Exponential Function .

Formula used:
Integration by parts
$\int {u \cdot vdx = u\int {vdx - \int {\left( {\dfrac{{du}}{{dx}}\int {vdx} } \right)} } } dx$
Here the criteria for selecting first function $u$ is according to the $ILATE$ rule , where $I$ denotes inverse Trigonometric Function, $L$ denotes Logarithm Function, $A$ denotes Algebraic Function, $T$ denotes Trigonometric Function, $E$ denotes Exponential Function .

Complete step by step solution:
We have to find the integration of function $x{\sec ^2}x$ ,
Therefore, we can write as follow ,
$I = \int {x \cdot {{\sec }^2}xdx}$ (to find)
For integrating this we have to use integration by parts that is
$\int {u \cdot vdx = u\int {vdx - \int {\left( {\dfrac{{du}}{{dx}}\int {vdx} } \right)} } } dx$
Here the criteria for selecting first function $u$ is according to the $ILATE$ rule , where $I$ denotes inverse Trigonometric Function, $L$ denotes Logarithm Function, $A$ denotes Algebraic Function, $T$ denotes Trigonometric Function, $E$ denotes Exponential Function .
Here $x$ is an algebraic function and ${\sec ^2}x$ is a trigonometric function. Algebraic function comes first.
Therefore, we take $x$ as our first function.
$\int {x \cdot {{\sec }^2}xdx = x\int {{{\sec }^2}xdx - \int {\left( {\dfrac{{dx}}{{dx}}\int {{{\sec }^2}x} } \right)dx} } }$
We know that integration of ${\sec ^2}x$ is equal to $\tan x$ ,
$= x\tan x - \int {1 \cdot \tan xdx}$
We also know that integration of $\tan x$ is equal to $\log \sec x$ ,
$= x\tan x - \log \sec x + c$
Hence , we have the required result.

Note: When doing indefinite integration, always write $c$ part after the integration. This $c$ part indicates the constant part remains after integration and can be understood when you explore it graphically. Infinite integration constant gets cancelled out, so we only write it in indefinite integration.