# Any idea how to solve this integral?

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.

• First step is probably u-sub. Nov 28, 2016 at 14:50

$$A=\int \frac{-u^{^{\frac{3}{2}}}\sin\left(\frac{\sqrt{11}}{2}ln\left(u\right)\right)}{\frac{\sqrt{11}}{2}\:ln\:\left(u\right)}du$$
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 $$B+iA=\int \frac{-u^{^{\frac{3}{2}}}e^{i\left(\frac{\sqrt{11}}{2}ln\left(u\right)\right)}}{\frac{\sqrt{11}}{2}\:ln\:\left(u\right)}du\\=\int {-u^{3/2}\cdot u^{i{\sqrt {11}\over 2}}\over {\sqrt{11}\over 2}\ln u}du$$