What is the integral of 20(sqrt(x^2-100))/(x^3) ?

The integral of 20(sqrt(x^2-100))/(x^3) is -(10sqrt(x^2-100))/(x^2)+arcsec(1/10 x)+C

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integral of 20(sqrt(x^2-100))/(x^3)

\int 20 \frac{x ^{2} - 100}{x ^{3}} d x

- \frac{10 x ^{2} - 100}{x ^{2}} + arcsec (\frac{1}{10} x) + C

Solution steps

\int 20 \cdot \frac{x ^{2} - 100}{x ^{3}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 20 \cdot \int \frac{x ^{2} - 100}{x ^{3}} d x

Apply Integration By Parts

: - \frac{x ^{2} - 100}{2 x ^{2}} - \int - \frac{1}{2 x x ^{2} - 100} d x

= 20 (- \frac{x ^{2} - 100}{2 x ^{2}} - \int - \frac{1}{2 x x ^{2} - 100} d x)

\int - \frac{1}{2 x x ^{2} - 100} d x = - \frac{1}{20} arcsec (\frac{1}{10} x)

= 20 (- \frac{x ^{2} - 100}{2 x ^{2}} - (- \frac{1}{20} arcsec (\frac{1}{10} x)))

Simplify

20 (- \frac{x ^{2} - 100}{2 x ^{2}} - (- \frac{1}{20} arcsec (\frac{1}{10} x))) : - \frac{10 x ^{2} - 100}{x ^{2}} + arcsec (\frac{1}{10} x)

= - \frac{10 x ^{2} - 100}{x ^{2}} + arcsec (\frac{1}{10} x)

Add a constant to the solution

= - \frac{10 x ^{2} - 100}{x ^{2}} + arcsec (\frac{1}{10} x) + C

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What is the integral of 20(sqrt(x^2-100))/(x^3) ?
The integral of 20(sqrt(x^2-100))/(x^3) is -(10sqrt(x^2-100))/(x^2)+arcsec(1/10 x)+C