Question

Find the integrals (primitives):
$\rm \displaystyle\int \Big( 3\sec x^{2} -\frac{4}{x} +\frac{1}{x\sqrt{x}}-7 \Big) dx$

Answer 1

To find the integration of

$\int \Big( 3\sec x^{2} -\frac{4}{x} +\frac{1}{x\sqrt{x}}-7 \Big) dx \\ \\ 3\int \: \sec x^{2} \: dx - 4\int \frac{1}{x} dx + \int {x}^{ \frac{ - 3}{2} } dx - 7\int \: 1dx \\ \\ = 3 \: tan \: x - 4 \: log \: x + \frac{ {x}^{ \frac{ - 3}{2} + 1} }{ \frac{ - 3}{2} + 1 } - 7x + c \\ \\ = 3 \: tan \: x - 4 \: log \: x + \frac{ {x}^{ \frac{ - 1}{2}} }{ \frac{ - 1}{2} } - 7x + C\\ \\So\\\\ \int \Big( 3\sec x^{2} -\frac{4}{x} +\frac{1}{x\sqrt{x}}-7 \Big) dx \\ \\ = 3 \: tan \: x - 4 \: log \: x - \frac{2}{ \sqrt{x} } - 7x + C$
Hope it helps you.