Answered: e* csc (3e + 4) dx

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.2: Exponents And Radicals

Problem 31E

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Question

Integration review Evaluate the following integrals.

Expert Solution

Step 1

Consider the given integration as

$\int e^{x} c s c (3 e^{x} + 4) d x$

Step 2

Substitute $t = 3 e^{x} + 4$ then differentiate as

$d t = 3 e^{x} d x$ then,

$\begin{array}{rcl} \int e^{x} c s c (3 e^{x} + 4) d x & = & \frac{1}{3} \int csc t d t \\ = & \frac{1}{3} \int c s c t (\frac{c s c t - c o t t}{c s c t - c o t t}) d t \end{array}$

Substitute $c s c t - c o t t = u$ then differentiate

$d u = (- c s c t c o t t + c s c^{2} t) d t$ then,

$\begin{array}{rcl} \int e^{x} c s c (3 e^{x} + 4) d x & = & \int \frac{d t}{u} \\ = & \ln |u| + C \end{array}$

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