Question

Find : $\int \frac{(2x-5)e^{2x}}{{(2x-3)}^3}dx.$

Answer 1

$\int \frac{(2x-5)e^{2x}}{{(2x-3)}^3}dx$

Put $2x=t$ , $dx=\frac{dt}{2}$ , we get $\int \frac{(t-5)e^t}{{(t-3)}^3}\frac{dt}{2}$

Now observe that $\frac{(t-5)}{{(t-3)}^3}=\frac{1}{{(t-3)}^2}-\frac{2}{{(t-3)}^3}$

so we get $(\frac{1}{{(t-3)}^2}-\frac{2}{{(t-3)}^3})e^t\frac{dt}{2}$

if we consider $f\left(t\right)=\frac{1}{{(t-3)}^2}$ , then $f^{{}^{\prime }}\left(t\right)=-\frac{2}{{(t-3)}^3}$

So the given problem is reduced to $\int \left(f\right(t)+f^{{}^{\prime }}\left(t\right))e^t\frac{dt}{2}=f\left(t\right)\frac{e^t}{2}+c=\frac{1}{{(t-3)}^2}\frac{e^t}{2}+c$ , now substitute $t=2x$ , we get $\frac{1}{{(2x-3)}^2}\frac{e^{2x}}{2}+c$