Chapter 5: Q. 45 (page 496)

Use whatever method you like to solve each of the given definite and indefinite integrals. These integrals are neither in order of difficulty nor in order of technique. Many of the integrals can be solved in more than one way.
$\int \frac{\sinh^{- 1} x}{\sqrt{x^{2} + 1}} d x$

Short Answer

Expert verified

The result is $\frac{{(\sinh^{- 1} x)}^{2}}{2} + C$ .

Step by step solution

Step 1. Given information.

Consider the given integral,

$\int \frac{\sinh^{- 1} x}{\sqrt{x^{2} + 1}} d x$

Step 2. Solve the integral.

As the denominator is the derivative of the numerator, apply the substitution method to find the integral.

$\begin{array}{rcl} \int \frac{\sinh^{- 1} x}{\sqrt{x^{2} + 1}} d x & = & \int \frac{\sinh^{- 1} x}{\sqrt{x^{2} + 1}} d x [\sinh^{- 1} x = t, \frac{1}{\sqrt{x^{2} + 1}} d x = d t] \\ = & \int t d t \\ = & \frac{t^{2}}{2} + c \\ = & \frac{{(\sinh^{- 1} x)}^{2}}{2} + C \end{array}$

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Use whatever method you like to solve each of the given definite and indefinite integrals. These integrals are neither in order of difficulty nor in order of technique. Many of the integrals can be solved in more than one way.
$\int \frac{\sinh^{- 1} x}{\sqrt{x^{2} + 1}} d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integral.

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Use whatever method you like to solve each of the given definite and indefinite integrals. These integrals are neither in order of difficulty nor in order of technique. Many of the integrals can be solved in more than one way. ∫sinh-1xx2+1dx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Logic and Functions

Statistics

Pure Maths

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Use whatever method you like to solve each of the given definite and indefinite integrals. These integrals are neither in order of difficulty nor in order of technique. Many of the integrals can be solved in more than one way.
$\int \frac{\sinh^{- 1} x}{\sqrt{x^{2} + 1}} d x$