Evaluate the integral. dxe

We have to evaluate the indefinite integral: ∫dxex1-e-2x Rearranging the integral,…

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College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter0: A Review Of Basic Algebra

Section0.4: Polynomials

Problem 129E

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Question

Evaluate the integral.

Expert Solution

Step 1

We have to evaluate the indefinite integral:

$\int \frac{d x}{e^{x} \sqrt{1 - e^{- 2 x}}}$

Rearranging the integral,

$\int \frac{d x}{e^{x} \sqrt{1 - e^{- 2 x}}} = \int \frac{d x}{e^{x} \sqrt{1 - {(e^{- x})}^{2}}}$

Now solving this by substitution method since derivative of function is present in the integral

so assuming, $t = e^{- x}$

differentiating,

$\begin{array}{rcl} \frac{d t}{d x} & = & \frac{d e^{- x}}{d x} \\ = & e^{- x} \frac{d (- x)}{d x} \\ = & e^{- x} (- \frac{d x}{d x}) \\ = & e^{- x} (- 1) \\ \frac{d t}{d x} & = & - e^{- x} \\ - d t & = & e^{- x} d x \\ = & \frac{1}{e^{x}} d x \\ = & \frac{d x}{e^{x}} \end{array}$

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