Q. 60 - Calculus

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: $\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$).

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

Step-by-Step Solution

Answer

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

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.

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