Question

Integrate the function: $\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}$ [Hint:$\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}$=$\frac{1}{x^{\frac{1}{3}(1+x^{\frac{1}{6}})}}$Put x=t6]

Answer 1

$\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}$=$\frac{1}{x^{\frac{1}{3}(1+x^{\frac{1}{6}})}}$

Let x=t6⇒dx=6t5dt

∴$\int \frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}dx=\frac{1}{x^{\frac{1}{3}(1+x^{\frac{1}{6}})}}dx$

=∫6t5/t2(1+t)dt

=6∫$\frac{t^3}{1+t}dt$