I'm solving an inhomogeneous differential equation and in the very last step, I must solve these two integrals:
$$\int\frac{-u^{^{\frac{3}{2}}}\sin\left(\dfrac{\sqrt{11}}{2}\ln u\right)}{\dfrac{\sqrt{11}}{2}\ln u}~du$$
$$\int\frac{u^{^{\frac{3}{2}}}\cos\left(\dfrac{\sqrt{11}}{2}\ln u\right)}{\dfrac{\sqrt{11}}{2}\ln u}~du$$
Since the procedure to solve them should be the same I would like to know if any of you guys can give some hint about how to solve them.
Thanks in advance.
I think this might help
$$A=\int \frac{-u^{^{\frac{3}{2}}}\sin\left(\frac{\sqrt{11}}{2}ln\left(u\right)\right)}{\frac{\sqrt{11}}{2}\:ln\:\left(u\right)}du$$
and $$B=\int \frac{-u^{^{\frac{3}{2}}}\cos\left(\frac{\sqrt{11}}{2}ln\left(u\right)\right)}{\frac{\sqrt{11}}{2}\:ln\:\left(u\right)}du$$
Now $$B+iA=\int \frac{-u^{^{\frac{3}{2}}}e^{i\left(\frac{\sqrt{11}}{2}ln\left(u\right)\right)}}{\frac{\sqrt{11}}{2}\:ln\:\left(u\right)}du\\=\int {-u^{3/2}\cdot u^{i{\sqrt {11}\over 2}}\over {\sqrt{11}\over 2}\ln u}du$$
I think this should be an easy integration by parts.
After solving , just seperate out the real and complex parts!