To follow up:
2.) [MATH]I=\int_0^{\alpha} x^{\frac{1}{3}}-x^{-\frac{1}{3}}\,dx=\left[\frac{3}{4}x^{\frac{4}{3}}-\frac{3}{2}x^{\frac{2}{3}}\right]_0^{\alpha}=\frac{3\alpha^{\frac{2}{3}}}{4}\left(\alpha^{\frac{2}{3}}-2\right)[/MATH]
This is equivalent to choice (a).
3.) [MATH]I=\int_1^{\alpha}\frac{1}{4x-1}\,dx=\left[\frac{1}{4}\ln(4x-1)\right]_1^{\alpha}=\frac{1}{4}\ln\left(\frac{4\alpha-1}{3}\right)[/MATH]
This is equivalent to choice (d).