Question

Integrate :

$\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}$

Answer 1

$I=\int \frac{1}{x^{1/2}+n^{1/3}}dx$

Put $x=p^6$

$dx=6p^{5}dp$.

$I=\int \frac{6p^{5}dp}{p^3+p^2}=\int \frac{6p^{3}dp}{p+1}$

$p+1=t$

$dp=dt$

$=\int \frac{6(t-1{)}^{3}dt}{t}=6\int \frac{t^3-3t^2+3t-1}{t}dt$

$T=6\int [t^2-3t+3-\frac{1}{t}]dt$

$=6[\frac{t^3}{3}-\frac{3t^2}{2}+3t-(n|t\left)\right]+c$

$=6[\frac{(P.H{)}^3}{3}-\frac{3(P.H{)}^2}{2}+3(PH)-In|PH|]+c$

$\int \frac{dx}{x^{1/3}+x^{1/2}}=6[\frac{(x^{1/6}+1{)}^3}{3}-\frac{3(x^{1/6}+1{)}^2}{2}+3(x^{1/6}+1)-In(x^{1/6}+1\left)\right]+c$