Q. 31 - 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.  

$\int 4{\left(e^{x-3}\right)}^{2}dx$

Step-by-Step Solution

Answer

The solution of the integral is $2e^{2x-6}+C.$

Step 1. Given Information.

The given integral is $\int 4{\left(e^{x-3}\right)}^{2}dx.$

Step 2. Solve.

By solving the integral we get, 

$$\begin{gathered} \int 4{\left(e^{x-3}\right)}^{2}dx \\ =\int 4\left(e^{2x-6}\right)dx \\ =4\int e^{2x-6}dx \\ \mathrm{Use} \int e^x dx=e^x+C \\ =4\left(\frac{1}{2}\right)e^{2x-6}+C \\ =2e^{2x-6}+C \end{gathered}$$

Step 3. Verification.

To verify the answer we differentiate $2e^{2x-6}+C$ it.

On differentiating we get,

$$\begin{gathered} 2e^{2x-6}+C \\ =\frac{d}{dx}\left(2e^{2x-6}\right)+\frac{d}{dx}\left(C\right) \\ =4e^{2x-6}+0 \\ =4e^{2x-6} \end{gathered}$$

Hence proved.