Answer: Preliminary steps The following integrals require

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

ISBN: 9780321570567 2

Solution for problem 54E Chapter 7.4

Calculus: Early Transcendentals | 1st Edition

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Problem 54E

Preliminary steps

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals.

Step-by-Step Solution:

SOLUTION

Step 1

We need to solve $\int\limits_{\ }^{\ }\frac{dx}{{e}^{x}+{e}^{2x}}$

Let us use the substitution method here.

Let $u={e}^{x\ \ \ }\ \ \ \Rightarrow{}du={e}^{x}dx\ =udx$

Therefore $\int\limits_{\ }^{\ }\frac{dx}{{e}^{x}+{e}^{2x}}=\int\limits_{\ }^{\ }\frac{du}{u(u+{u}^{2})}=\int\limits_{\ }^{\ }\frac{du}{{u}^{3}+{u}^{2}}=\int\limits_{\ }^{\ }\frac{du}{{u}^{2}(u+1)}$