Question

Integrate the following functions.
$\int \frac{x}{\sqrt{x}+4}dx.$

Answer 1

$\int \frac{x}{\sqrt{x}+4}dx=\int \frac{x+4-4}{\sqrt{x+4}}dx=\int \frac{x+4}{\sqrt{x+4}}dx-4\int \frac{1}{\sqrt{x+4}}dx=\int (x+4{)}^{\frac{1}{2}}dx-4\int (x+4{)}^{-\frac{1}{2}}dx=\frac{(x+4{)}^{\left(\frac{1}{2}\right)+1}}{\left(\frac{1}{2}\right)+1}-4\frac{(x+4{)}^{(-\frac{1}{2})+1}}{(-\frac{1}{2})+1}+C=\frac{2}{3}(x+4{)}^{\frac{3}{2}}-8(x+4{)}^{\frac{1}{2}}+C$