Browse Questions

Evaluate the following $\large\int\frac{ x^{\frac{1}{2}}}{1+x^{\frac{3}{4}}}dx$ $(Hint:Put\;x=z^4)$

Toolbox:
• If $f(x)$ is substituted by t; then $f'(x)dx=f'(t)dt$ and $\int f(x)dx=\int f(t)dt$
• $\large\int \frac{dx}{(x+a)}=\log |x+a|+c$
Step 1:
Let $I=\large\int\frac{ x^{\frac{1}{2}}}{1+x^{\frac{3}{4}}}dx$
Put $x=z^4$
Differentiating with respect to z we get
$dx=4z^3 .dz$
On substituting for x and dx in I we get,
Therefore $I=\large\int \frac{(z^4)^{1/2}}{1+(z^4)^{3/4}}.4z^3dz$
$I=\large\int \frac{z^2}{1+z^3}.4 z^3.dz$