Question

Find the integrals (primitives):
$\rm \displaystyle\int \frac{1}{\sqrt{5x+3}-\sqrt{5x+1}} dx$

Answer 1

For integration of the given function ,first do rationalisation of denominator

$\frac{1}{\sqrt{5x+3}-\sqrt{5x+1}} \times \frac{\sqrt{5x+3} + \sqrt{5x+1}}{\sqrt{5x+3} + \sqrt{5x+1}} \\ \\ applying \: identity \: in \: denominator \\ \\ = \frac{\sqrt{5x+3} + \sqrt{5x+1}}{( { \sqrt{5x + 3} })^{2} - ( { \sqrt{5x + 1} })^{2} } \\ \\ = \frac{\sqrt{5x+3} + \sqrt{5x+1}}{5x + 3 - 5x - 1} \\ \\ \frac{\sqrt{5x+3}-\sqrt{5x+1}}{2}$
$\int \: \frac{ \sqrt{5x + 3} }{2} dx -\int \: \frac{ \sqrt{5x + 1} }{2} dx \\ \\ \frac{1}{2} \int \: \sqrt{5x + 3} \: dx - \frac{1}{2} \int \: \sqrt{5x + 1} \: dx \\ \\ applying \: power \: rule \:and \: coefficient \: rule \: of \: integration \\ \\ = \frac{ {(5x + 3)}^{( \frac{1}{2} + 1) } }{10( \frac{1}{2} + 1)} - \frac{ {(5x + 1)}^{( \frac{1}{2} + 1) } }{10( \frac{1}{2} + 1)} + c \\ \\ = \frac{ {(5x + 3)}^{( \frac{3}{2} ) } }{15} - \frac{ {(5x + 1)}^{( \frac{3}{2}) } }{15} + c \\ \\ = \frac{1}{15} \bigg({(5x + 3)}^{( \frac{3}{2} ) } - {(5x + 1)}^{( \frac{3}{2} ) }\bigg) + C \:$
$\int \frac{1}{\sqrt{5x+3}-\sqrt{5x+1}} dx \\\\= \frac{1}{15} \bigg[{(5x + 3)}^{( \frac{3}{2} ) } - {(5x + 1)}^{( \frac{3}{2} ) }\bigg] + C \:$