Mathematics - Indefinite Integration Question with Solution | TestHub

The integral can be written as $\int e^x\frac{x^2}{(x+2{)}^2}dx$. We can manipulate the term $\frac{x^2}{(x+2{)}^2}$ to fit the form $f(x)+f^{\prime }(x)$. We add and subtract 4 in the numerator: $\frac{x^2}{(x+2{)}^2}=\frac{x^2-4+4}{(x+2{)}^2}=\frac{(x-2)(x+2)+4}{(x+2{)}^2}=\frac{x-2}{x+2}+\frac{4}{(x+2{)}^2}$. Let $f(x)=\frac{x-2}{x+2}$. Then $f^{\prime }(x)=\frac{1(x+2)-(x-2)1}{(x+2{)}^2}=\frac{x+2-x+2}{(x+2{)}^2}=\frac{4}{(x+2{)}^2}$. So the integral is of the form $\int e^x(f(x)+f^{\prime }(x))dx=e^{x}f(x)+C$. Therefore, the integral is $e^x\left(\frac{x-2}{x+2}\right)+C$.