Q. 55 - 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: $\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$).

$\int (e^x\sqrt{x}+\frac{e^x}{2\sqrt{x}})dx.$

Step-by-Step Solution

Answer

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

Step 1. Given Information.

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

Step 2. Formula involved.

$\int e^x(u+\frac{du}{dx})dx = e^x\left(u\right) +c.$

Step 3. Solving the integral.

$$\begin{gathered} \int e^x(\sqrt{x}+\frac{1}{2\sqrt{x}})dx \\ = \int e^x(\sqrt{x}+\frac{d}{dx}(\sqrt{x}\left)\right)dx \\ from step 2 we get, \\ =e^x\sqrt{x}+c. \end{gathered}$$