Question

The value of $\int e^x(x^5+5x^4+1).dx$ is

• A. $e^x.x^5+e^x+C$
• B. $e^x.x^5$
• C. $5x^4.e^x$
• D. $e^{x+1}.x^5+C$

Answer 1

Answer: (A)

$e^x.x^5+e^x+C$

Answer 2

Let $I = \int e^{x} \left(x^{5} + 5x^{4} +1\right)dx$
$= \int e^{x} x^{5} dx + 5 \int e^{x} x^{4} dx + \int e^{x} dx$
$= x^{5} e^{x} - \int 5x^{4} e^{x} dx + 5 \int e^{x} x^{4} dx + e^{x}$
$= x^{5} e^{x} + e^{x} + c = e^{x} \left(x^{5} + 1\right)+ c$