Make a substitution to express the integrand as a rational function and then evaluate the...
Question:
Make a substitution to express the integrand as a rational function and then evaluate the integral. {eq}\int \frac{e^{2x}}{e^{2x}+3e^x+2}dx {/eq}
Partial Fraction Decomposition
To integrate the rational function {eq}f(x)=\frac{cx+d}{(x+a)(x+b)} {/eq} (we are assuming {eq}a\neq b {/eq}), we first need to decompose {eq}f(x) {/eq} into partial fractions
{eq}\displaystyle \frac{cx+d}{(x+a)(x+b)}=\frac{A}{x+a}+\frac{B}{x+b},\quad (\ast) {/eq}
where {eq}A {/eq} and {eq}B {/eq} are constants. To find {eq}A {/eq} and {eq}B {/eq}, we multiply both sides of {eq}(\ast) {/eq} by {eq}(x+a)(x+b) {/eq} and get
{eq}cx+d=A(x+b)+B(x+a)=(A+B)x+(Ab+Ba). {/eq}
Then, {eq}A {/eq} and {eq}B {/eq} can be found by solving the system
{eq}A+B=c,\quad Ab+Ba=d. {/eq}
Answer and Explanation: 1
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View this answerApplying the change of variable {eq}e^x=t {/eq}, we have {eq}e^xdx=dt {/eq}, so
{eq}\displaystyle \begin{align*} \int...
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How to Integrate Functions With Partial Fractions
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Chapter 13 / Lesson 10
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Learn about integration by partial fractions. Explore how to make partial fractions and then how to integrate fractions. See examples of integrating fractions.
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