Question

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: $\tan \left(x\right)=\frac{\sin \left(x\right)}{\cos \left(x\right)}$).

$\int \frac{1}{2\sqrt{x^2-1}}dx.$

Short answer

The value of the given integral is$-\frac{1}{2}{\sin }^{-1}\left(x\right)+c.$

Step-by-step solution

Step 1. Given Information.

Given is a integral:$\int \frac{1}{2\sqrt{x^2-1}}dx.$

Step 2. Formula involved.

$\int \frac{1}{\sqrt{1-x^2}}dx={\sin }^{-1}\left(x\right)+c.$

Step 3. Solving the integral.

$$\begin{gathered} \int \frac{1}{2\sqrt{x^2-1}}dx \\ =-\frac{1}{2}\int \frac{1}{\sqrt{1-x^2}}dx \\ =-\frac{1}{2}{\sin }^{-1}\left(x\right)+c. \end{gathered}$$