Question

Evaluate the following:

${\displaystyle \int \frac{x^{\frac{1}{2}}}{1+x^{\frac{3}{4}}}\text{d}x}$   (Hint: Put ${\displaystyle \sqrt{x}}$ = z⁴)

Answer 1

Let I = ${\displaystyle \int \frac{x^{\frac{1}{2}}}{1+x^{\frac{3}{4}}}\text{d}x}$ 

Put x = t⁴

⇒ dx = 4t³d dt

∴ I = ${\displaystyle 4\int \frac{{\text{t}}^2\left({\text{t}}^3\right)}{1+{\text{t}}^3}\text{dt}}$

= ${\displaystyle 4\int ({\text{t}}^2-\frac{{\text{t}}^2}{1+{\text{t}}^3})\text{dt}}$

= ${\displaystyle 4\int {\text{t}}^2\text{dt}-4\int {\text{t}}^2(1+{\text{t}}^3)\text{dt}}$

= ${\displaystyle 4\frac{{\text{t}}^3}{3}-\frac{4}{3}\int \frac{3{\text{t}}^2}{1+{\text{t}}^3}\text{dt}}$

= ${\displaystyle 4\frac{{\text{t}}^3}{3}-\frac{4}{3}\log |1+{\text{t}}^3|+\text{C}}$

= ${\displaystyle \frac{4}{3}{x}^{\frac{3}{4}}-\frac{4}{3}\log \left|(1+x^{\frac{3}{4}})\right|+\text{C}}$