Evaluate the indefinite integral \int \frac {8-3xe^x}{x} dx


Evaluate the indefinite integral {eq}\int \frac {8-3xe^x}{x} dx {/eq}

Indefinite Integral in Calculus:

The indefinite integration can be written as {eq}\int f(x) \ dx {/eq} . The symbol {eq}\int {/eq} represents integration, and {eq}dx {/eq} is a differential of the variable {eq}x {/eq} .

We can use many formulas and common integrals to solve integral problems. To integrate we may use the common integrals {eq}\displaystyle \int \dfrac{1}{x} \ dx = \ln|x |+C , \int e^x \ dx =e^x+C. {/eq}.

Answer and Explanation:

We are given:

{eq}\displaystyle \int \frac {8-3xe^x}{x} dx {/eq}

Break the fraction:

{eq}=\displaystyle \int \left( \dfrac{8}{x}+ \dfrac{ 3xe^x}{x} \right) \ dx {/eq}

Apply the integral sum rule:

{eq}=\displaystyle \int \dfrac{8}{x} \ dx - \int 3e^x \ dx {/eq}

Take the constant out:

{eq}= \displaystyle 8 \int \dfrac{1}{x} \ dx -3 \int e^x \ dx {/eq}

Apply common integrals:

{eq}= \displaystyle 8 \ln|x| -3 e^x +C {/eq}

Therefore, the solution is:

{eq}\displaystyle {\boxed{ \int \frac {8-3xe^x}{x} dx = \displaystyle 8 \ln|x| -3 e^x +C. }} {/eq}

Learn more about this topic:

Indefinite Integrals as Anti Derivatives

from Math 104: Calculus

Chapter 12 / Lesson 11

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