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.

  • 1
    $\begingroup$ First step is probably u-sub. $\endgroup$ Nov 28 '16 at 14:50

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!


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