Short Answer

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: $\tan (x) = \frac{\sin (x)}{\cos (x)}$ ).
$\int \frac{e^{3 x} - 2 e^{4 x}}{e^{2 x}} d x$ .

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

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TABLE OF CONTENTS

Step 1. Given Information.

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

Step 2. Formula involved.

$\int e^{x} d x = e^{x} + c .$ and $\int e^{a x} d x = \frac{e^{a x}}{a} + c, w h e r e a i s a c o n s \tan t .$

Step 3. Solving the integral.

$\int \frac{e^{3 x} - 2 e^{4 x}}{e^{2 x}} d x = \int (e^{3 x - 2 x} - 2 e^{4 x - 2 x}) d x = \int (e^{x} - 2 e^{2 x}) d x = \int e^{x} d x - \int 2 e^{2 x} d x = e^{x} - 2 \frac{e^{2 x}}{2} + c = e^{x} - e^{2 x} + c .$

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