Help with integration using a given integral

[Hao Do]

Can anyone please help me with these 2 questions?
For b, I attempted to solve it by using integration by parts. But it seemed like this way didn’t let me incorporate the given integral.
For c, I don’t have any idea at all,
Thank you for reading my questions

 

[HallsofIvy]

No "integration by parts", just substitution.

For (b) I would use the substitution \(\displaystyle u= 2x\). Then \(\displaystyle du= 2dx\) so that \(\displaystyle dx= \frac{1}{2}du\) and \(\displaystyle x= \frac{u}{2}\) so \(\displaystyle x^3= \frac{u^3}{8}\).

For (c) I would use the substitution \(\displaystyle u= \frac{x- 1}{\sqrt{2}}\).