Copyright

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

Question:

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:

Loading...
Indefinite Integrals as Anti Derivatives

from Math 104: Calculus

Chapter 12 / Lesson 11
7.4K

Related to this Question

Explore our homework questions and answers library