evaluate the following indefinite integral. \n int \frac{x^{4} e^{x}-3 x^{3}}{x^{4}} d x \n int \frac{x^{4}…

evaluate the following indefinite integral. \n int \frac{x^{4} e^{x}-3 x^{3}}{x^{4}} d x \n int \frac{x^{4} e^{x}-3 x^{3}}{x^{4}} d x=quad \n(use parentheses to clearly denote the argument of each function.)

evaluate the following indefinite integral. \n int \frac{x^{4} e^{x}-3 x^{3}}{x^{4}} d x \n int \frac{x^{4} e^{x}-3 x^{3}}{x^{4}} d x=quad \n(use parentheses to clearly denote the argument of each function.)

Answer

Explanation:

Step1: Simplify the integrand

Divide each term in the numerator by (x^{4}): [ \begin{align*} \int\frac{x^{4}e^{x}-3x^{3}}{x^{4}}dx&=\int\left(\frac{x^{4}e^{x}}{x^{4}}-\frac{3x^{3}}{x^{4}}\right)dx\ &=\int\left(e^{x}-\frac{3}{x}\right)dx \end{align*} ]

Step2: Integrate term - by - term

Use the integral rules (\int e^{x}dx = e^{x}+C_1) and (\int\frac{a}{x}dx=a\ln|x| + C_2) ((a = 3) in our case): [ \begin{align*} \int\left(e^{x}-\frac{3}{x}\right)dx&=\int e^{x}dx-3\int\frac{1}{x}dx\ &=e^{x}-3\ln|x|+C \end{align*} ]

Answer:

(e^{x}-3\ln|x| + C)