Question

Find the integrals (primitives):
$\rm \displaystyle\int \Big(\sqrt{5x-1}-\frac{1}{\sqrt{5x+2}}\Big) \ dx$

Answer 1

We know that

$\int \: {x}^{n} dx = \frac{ {x}^{n + 1} }{n + 1} + C$
$\int \Big(\sqrt{5x-1}-\frac{1}{\sqrt{5x+2}}\Big) \ dx \\ \\ \int \sqrt{5x-1} \: dx - \int\frac{1}{\sqrt{5x+2}} \: dx \\ \\ \int \: ( {5x - 1})^{ \frac{1}{2} } dx - \int \: ( {5x + 2)}^{ \frac{ - 1}{2} } dx$
Apply power rule

$= \frac{ {(5x - 1)}^{ \frac{1}{2} + 1 } }{5( \frac{1}{2} + 1) } - \frac{ {(5x +2)}^{ \frac{ - 1}{2} + 1 } }{5( \frac{ - 1}{2} + 1) } + C \\ \\ = \frac{ {(5x - 1)}^{ \frac{3}{2} } }{5( \frac{3}{2}) } - \frac{ {(5x +2)}^{ \frac{ 1}{2} } }{5( \frac{1}{2}) } + C \\ \\ \int \Big(\sqrt{5x-1}-\frac{1}{\sqrt{5x+2}}\Big) \ dx \\\\= \frac{2{(5x - 1)}^{ \frac{3}{2} }}{15} - \frac{{2(5x +2)}^{ \frac{1}{2} }}{5} + C$
Hope it helps you.