Question

Find the indefinite integral

Answer 1

Answer: $\displaystyle \frac{2}{3}x^{3/2} + \frac{2}{5}x^{1/2}+C$

This is equivalent to $\frac{2}{3}\sqrt{x^3} + \frac{2}{5}\sqrt{x}+C$

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Work Shown:

$\displaystyle \int\left(\sqrt{x} + \frac{1}{5\sqrt{x}}\right)dx\\\\\\ \displaystyle \int\left(\sqrt{x}\right)dx + \int\left(\frac{1}{5\sqrt{x}}\right)dx\\\\\\ \displaystyle \int\left(x^{1/2}\right)dx + \int\left(\frac{1}{5}x^{-1/2}\right)dx\\\\\\ \displaystyle \int\left(x^{1/2}\right)dx + \frac{1}{5}\int\left(x^{-1/2}\right)dx$

$\displaystyle \frac{1}{1+1/2}x^{1+1/2} + \frac{1}{5}*\frac{1}{1+(-1/2)}x^{1+(-1/2)}+C\\\\\\ \displaystyle \frac{1}{3/2}x^{3/2} + \frac{1}{5}*\frac{1}{1/2}x^{1/2}+C\\\\\\ \displaystyle \frac{2}{3}x^{3/2} + \frac{1}{5}*2x^{1/2}+C\\\\\\ \displaystyle \frac{2}{3}x^{3/2} + \frac{2}{5}x^{1/2}+C$