(a) Find: ∫ (d x)/(x^1 / 2+ x^1 / 3)

$\displaystyle \int \dfrac{dx}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}=6 \int \dfrac{t^{5}}{t^{3}+t^{2}} dt=6 \int \dfrac{t^{3}}{t+1} dt$

$\displaystyle =6 \int\left(t^{2}-t+1-\dfrac{1}{t+1}\right) dt$

$\displaystyle =6\left(\dfrac{t^{3}}{3}-\dfrac{t^{2}}{2}+t-\log |t+1|\right)+c$

$\displaystyle =2 x^{\frac{1}{2}}-3 x^{\frac{1}{3}}+6 x^{\frac{1}{6}}-6 \log\left|x^{\frac{1}{6}}+1\right|+c$

Getting $\displaystyle \int\left(\dfrac{\pi}{4}-\tan^{-1} x\right) dx$

$\displaystyle =\dfrac{\pi x}{4}-\left[x \tan^{-1} x-\int \dfrac{x}{1+x^{2}} dx\right]$

$\displaystyle =\dfrac{\pi x}{4}-x \tan^{-1} x+\dfrac{1}{2} \log\left(1+x^{2}\right)+c$