Q. 49 - 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: $\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$).

$\int \frac{e^{3x}-2e^{4x}}{e^{2x}}dx$.

Step-by-Step Solution

Answer

The value of the given integral will be $e^x-e^{2x}+c$.

Step 1. Given Information.

Given the integral: $\int \frac{e^{3x}-2e^{4x}}{e^{2x}}dx$.

Step 2. Formula involved.

$\int e^{x}dx = e^x+c.$ and $\int e^{ax}dx = \frac{e^{ax}}{a}+c, where a is a cons\tan t.$

Step 3. Solving the integral.

$$\begin{gathered} \int \frac{e^{3x}-2e^{4x}}{e^{2x}}dx \\ =\int (e^{3x-2x}-2e^{4x-2x})dx \\ =\int (e^x-2e^{2x})dx \\ =\int e^{x}dx - \int 2e^{2x}dx \\ =e^x - 2\frac{e^{2x}}{2} +c \\ =e^x - e^{2x} +c. \end{gathered}$$