Find indefinite integral $\int \frac{e^x(x^4+2)}{(1+x^2)^{5/2}}dx$

Question

Although I looked up the answer on integral calculator com but I still have little to no idea as to how one would proceed to solve this integral. Integrate $\dfrac{e^x(x^4+2)}{(1+x^2)^{5/2}}$ wrt $x.$ I initially tried to convert it to the form $e^x\cdot(f(x)+f'(x)).$ However, I wasn't successful in spite of struggling for more than half an hour :(

Any help would be appreciated :)

Answer 1

Hint

$$\dfrac{e^x\cdot f(x)}{dx}=?$$

So, it is sufficient to find $f(x)$ such that $$f(x)+f'(x)=\dfrac{x^4+2}{(1+x^2)^{5/2}}$$

As the degree of the denominator is $(5/2)\cdot2$

We can safely start with $$f(x)=\dfrac{a_0+a_1x+a_2x^2+a_3x^3+a_4x^4}{(1+x^2)^{5/2-1}}$$

Answer 2

Note

$$\left[\frac x{(1+x^2)^{3/2}}\right]’ =\frac {1-2x^2}{(1+x^2)^{5/2}} =\frac {x^4+2}{(1+x^2)^{5/2}}-\frac1{(1+x^2)^{1/2}}$$

Then, integrate by parts

$$\int \frac{e^x(x^4+2)}{(1+x^2)^{5/2}}dx = \int e^x d\left[\frac x{(1+x^2)^{3/2}}\right] +\int \frac{e^x}{(1+x^2)^{1/2}}dx$$ $$=\frac {e^xx}{(1+x^2)^{3/2}} +\int \left[ -\frac{xe^x}{(1+x^2)^{3/2}} +\frac{e^x}{(1+x^2)^{1/2}}\right]dx$$ $$=\frac {e^x x}{(1+x^2)^{3/2}} +\frac{e^x}{(1+x^2)^{1/2}}+C $$