Q. 56 - 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: $\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$).

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

Step-by-Step Solution

Answer

The value of the integral is $e^{\sqrt{x}}+c.$

Step 1. Given Information.

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

Step 2. Formula involved.

$\int e^{x}dx = e^x+c.$

Step 3. Solving the integral.

$$\begin{gathered} \int \frac{1}{2\sqrt{x}}{e}^{\sqrt{x}}dx \\ Let t = \sqrt{x} \\ dt = \frac{1}{2\sqrt{x}}dx \\ Putting its value in the integral we get, \\ = \int e^{t}dt \\ = e^t+c = e^{\sqrt{x}}+c. \end{gathered}$$