Question

How do you integrate $ \int {x{{\sec }^{ - 1}}} \left( x \right)dx $ ?

Answer 1

Hint:According to the question, we have to find the integration of $ \int {x{{\sec }^{ - 1}}} \left( x \right)dx $ .
So, first of all we have to integrate the given function by using the integration parts formula that is mentioned below.
Integration by parts formula: Integration by parts formula is used for integrating the product of two functions. This method is used to find the integrals by reducing them into standard forms. For example, if we have to find the integration of $ x\sin x $ , then we need to use this formula. The integrand is the product of the two functions. The formula for integrating by parts is given by;
 $ \Rightarrow \int {uvdx = u\int {vdx} } - \int {\left[ {\dfrac{{du}}{{dx}}\int {vdx} } \right]} dx............................(A) $
Now, we have to use the formula for differentiation of $ {\sec ^{ - 1}}x $ and integration of $ {x^n} $ as mentioned below,
 $ \Rightarrow \dfrac{d}{{dx}}\left( {{{\sec }^{ - 1}}x} \right) = \dfrac{1}{{x\sqrt {{x^2} - 1} }}...........................(B) $
 $ \Rightarrow \int {{x^n}dx} = \dfrac{{{x^{n + 1}}}}{{n + 1}}...........................(C) $
Now, we have to let $ {x^2} - 1 $ = t and convert the integrated function into $ dt $ form and solve the function.

Complete step by step answer:
Step1: First of all we have to integrate the given function by using integration parts formula (A) that is mentioned in the solution hint.
\[ \Rightarrow \int {x{{\sec }^{ - 1}}\left( x \right)dx = {{\sec }^{ - 1}}\left( x \right)\int {xdx} } - \int {\left[ {\dfrac{d}{{dx}}\left( {{{\sec }^{ - 1}}x} \right)\int {xdx} } \right]} dx\]
Step 2: Now, we have to use the formula (B) and (C) for differentiation of $ {\sec ^{ - 1}}x $ and integration of $ {x^n} $ respectively as mentioned in the solution step,
\[ \Rightarrow {\sec ^{ - 1}}\left( x \right) \times \dfrac{{{x^{1 + 1}}}}{{1 + 1}} - \int {\left[ {\dfrac{1}{{x\sqrt {{x^2} - 1} }} \times \dfrac{{{x^{1 + 1}}}}{{1 + 1}}} \right]} dx\]
\[ \Rightarrow {\sec ^{ - 1}}\left( x \right) \times \dfrac{{{x^2}}}{2} - \int {\left[ {\dfrac{1}{{x\sqrt {{x^2} - 1} }} \times \dfrac{{{x^2}}}{2}} \right]} dx\]
Now, we have to simplify the above expression by eliminating the term $ x. $
\[ \Rightarrow {\sec ^{ - 1}}\left( x \right) \times \dfrac{{{x^2}}}{2} - \int {\left[ {\dfrac{1}{{\sqrt {{x^2} - 1} }} \times \dfrac{x}{2}} \right]} dx\]
Step 3: Now, we have to let $ {x^2} - 1 $ = t and convert the integrated function into $ dt $ form.
 $ \Rightarrow {x^2} - 1 = t $
Now, we have to differentiate the above expression with respect to $ x $ both sides,
 $ \Rightarrow \dfrac{d}{{dx}}{x^2} - \dfrac{d}{{dx}}1 = \dfrac{d}{{dx}}t $
Now, we have to know that differentiation of $ {x^2} $ is $ 2x $ and differentiation of constant term is 0,
 $
   \Rightarrow 2x - 0 = \dfrac{{dt}}{{dx}} \\
   \Rightarrow 2xdx = dt \\
   \Rightarrow xdx = \dfrac{{dt}}{2} \\
  $
Step 4: Now, we have to put all the values of $ {x^2} - 1 $ and $ xdx $ from the solution step 3 to the expression obtained in the solution step 2.
\[
   \Rightarrow {\sec ^{ - 1}}\left( x \right) \times \dfrac{{{x^2}}}{2} - \int {\left[ {\dfrac{1}{{\sqrt t }} \times \dfrac{{dt}}{{2 \times 2}}} \right]} \\
   \Rightarrow {\sec ^{ - 1}}\left( x \right) \times \dfrac{{{x^2}}}{2} - \dfrac{1}{4}\int {{t^{ - 1/2}}} dt \\
 \]
Step 5: Now, we have to integrate the above integrated function as obtained in the solution step 4 with the help of the formula (C) as mentioned in the solution hint.
\[
   \Rightarrow {\sec ^{ - 1}}\left( x \right) \times \dfrac{{{x^2}}}{2} - \dfrac{1}{4} \times \dfrac{{{t^{ - 1/2 + 1}}}}{{ - 1/2 + 1}} \\
   \Rightarrow {\sec ^{ - 1}}\left( x \right) \times \dfrac{{{x^2}}}{2} - \dfrac{1}{4} \times \dfrac{{{t^{1/2}}}}{{1/2}} \\
   \Rightarrow {\sec ^{ - 1}}\left( x \right) \times \dfrac{{{x^2}}}{2} - \dfrac{1}{4} \times 2 \times {t^{1/2}} \\
 \]
Now, we have to put the value of t as $ {x^2} - 1 $ in the expression obtain just above,
\[ \Rightarrow {\sec ^{ - 1}}\left( x \right) \times \dfrac{{{x^2}}}{2} - \dfrac{1}{2} \times \sqrt {{x^2} - 1} \]
Final solution: Hence, the integration of $ \int {x{{\sec }^{ - 1}}} \left( x \right)dx $ is\[{\sec ^{ - 1}}\left( x \right) \times \dfrac{{{x^2}}}{2} - \dfrac{1}{2} \times \sqrt {{x^2} - 1} \].

Note:
It is necessary to understand about the Integration by parts formula as mentioned in the solution hint.
It is necessary to let $ {x^2} - 1 $ = t and convert the integrated function into $ dt $ form and solve the function.