Integrate $x^{3}e^{x}\int\frac{x+2}{x^{3}e^{x}}dx$

Question

Integrate $$x^{3}e^{x}\int\frac{x+2}{x^{3}e^{x}}dx$$

My attempt: $$ x^{3}e^{x} \int \frac{1}{x^{2}e^{x}} + \frac{2}{x^{3}e^{x}}dx = x^{3}e^{x} \big[ \int \frac{2}{x^{3}e^{x}}dx+ x^{-2}(-e^{-x}) - \int -2x^{-3}(-e^{-x})dx \big] $$ I integrate by parts for the term $\int \frac{1}{x^2e^{x}}dx$

Then $$ = x^{3}e^{x} \big[ \int \frac{2}{x^{3}e^{x}}dx - \frac{1}{x^{2}e^{x}} - \int \frac{2}{x^{3}e^{x}}dx \big] =\frac{-x^{3}e^{x}}{x^{2}e^{x}} = -x $$

So, my answer is $-x$. But the answer given includes a constant: $-x + Cx^{3}e^{x}$.

Is there something wrong with my answer?

Answer 1

Be careful, there is a constant of integration ($C$) inside the bracket. So you first add $C$.

Answer 2

If $\int{f}(x)=F(x)$ and $\int{f}(x)=G(x)$, then can you say $\int{(f(x)-g(x))}=F(x)-G(x)=0$? Of course not.... Consider $f(x)=2sinxcosx$ and $g(x)=sin2x$ and integrate first one with integration by parts. See how the constant plays out...