What is the integral of (sqrt(16x^2-1))/(x^3) ?

The integral of (sqrt(16x^2-1))/(x^3) is -(sqrt(16x^2-1))/(2x^2)+8arctan(sqrt(16x^2-1))+C

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integral of (sqrt(16x^2-1))/(x^3)

\int \frac{16 x ^{2} - 1}{x ^{3}} d x

- \frac{16 x ^{2} - 1}{2 x ^{2}} + 8 arctan (16 x^{2} - 1) + C

Solution steps

\int \frac{16 x ^{2} - 1}{x ^{3}} d x

Apply Integration By Parts

: - \frac{16 x ^{2} - 1}{2 x ^{2}} - \int - \frac{8}{x 16 x ^{2} - 1} d x

= - \frac{16 x ^{2} - 1}{2 x ^{2}} - \int - \frac{8}{x 16 x ^{2} - 1} d x

\int - \frac{8}{x 16 x ^{2} - 1} d x = - 8 arctan (16 x^{2} - 1)

= - \frac{16 x ^{2} - 1}{2 x ^{2}} - (- 8 arctan (16 x^{2} - 1))

Simplify

= - \frac{16 x ^{2} - 1}{2 x ^{2}} + 8 arctan (16 x^{2} - 1)

Add a constant to the solution

= - \frac{16 x ^{2} - 1}{2 x ^{2}} + 8 arctan (16 x^{2} - 1) + C

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What is the integral of (sqrt(16x^2-1))/(x^3) ?
The integral of (sqrt(16x^2-1))/(x^3) is -(sqrt(16x^2-1))/(2x^2)+8arctan(sqrt(16x^2-1))+C