Evaluate the following integrals:

Question:

Evaluate: $\int(1-\mathrm{x}) \sqrt{\mathrm{x}} \mathrm{dx}$.

Solution:

Given, $\int(1-x) \sqrt{x} \mathrm{dx}$

$=\int(\sqrt{x}-x \sqrt{x}) d x$

$=\int\left(x^{\frac{1}{2}}-x \cdot x^{\frac{1}{2}}\right) d x$

$=\int x^{\frac{1}{2}}-x^{\frac{3}{2}} d x$

$=\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}-\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}+c\left[\right.$ since, $\left.\int x^{n} d x=\frac{x^{n+1}}{n+1}\right]$

$=\frac{x^{\frac{3}{2}}}{\frac{3}{2}}-\frac{x^{\frac{5}{2}}}{\frac{5}{2}}+c$

$=\frac{2}{3} x^{\frac{3}{2}}-\frac{2}{5} x^{\frac{5}{2}}+c$

Leave a comment

Close

Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now