Question

Evaluate $\int \frac{x+9}{(x+10{)}^2}{e}^{x}dx=$

• A. $e^x\frac{1}{x+9}+c$
• B. $e^x\frac{1}{x+10}+c$
• C. $e^x\frac{1}{(x+9{)}^2}+c$
• D. $e^x+c\frac{1}{(x+10{)}^2}$

Answer 1

Answer: (B)

$e^x\frac{1}{x+10}+c$

$\int e^x\frac{x+9}{(x+10{)}^2}dx$

$=\int e^x\frac{\left[\right(x+10)-1]}{(x+10{)}^2}dx$

$=\int e^x[\frac{1}{(x+10)}-\frac{1}{(x+10{)}^2}]dx$

It is in the form of

$\int e^x\left(f\right(x)+f^{{}^{\prime }}(x\left)\right)dx$

$=e^{x}f\left(x\right)+c$

$\Rightarrow \int e^x[\frac{1}{(x+10)}-\frac{1}{(x+10{)}^2}]dx$

$=\frac{e^x}{(x+10)}+c$

$\therefore \int \frac{x+9}{(x+10{)}^2}{e}^{x}dx=\frac{e^x}{(x+10)}+c$