What is the integral of (5sqrt(x^2-1))/(x^2) ?

The integral of (5sqrt(x^2-1))/(x^2) is 5(-(sqrt(x^2-1))/x+ln|sqrt(x^2-1)+x|)+C

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integral of (5sqrt(x^2-1))/(x^2)

\int \frac{5 x ^{2} - 1}{x ^{2}} d x

5 (- \frac{x ^{2} - 1}{x} + ln x^{2} - 1 + x) + C

Solution steps

\int \frac{5 x ^{2} - 1}{x ^{2}} d x

Take the constant out:

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

= 5 \cdot \int \frac{x ^{2} - 1}{x ^{2}} d x

Apply Integration By Parts

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

= 5 (- \frac{x ^{2} - 1}{x} - \int - \frac{1}{x ^{2} - 1} d x)

\int - \frac{1}{x ^{2} - 1} d x = - ln x^{2} - 1 + x

= 5 (- \frac{x ^{2} - 1}{x} - (- ln x^{2} - 1 + x))

Simplify

= 5 (- \frac{x ^{2} - 1}{x} + ln x^{2} - 1 + x)

Add a constant to the solution

= 5 (- \frac{x ^{2} - 1}{x} + ln x^{2} - 1 + x) + C

y = 5 x - x^{2}, (1, 4)

y \to 1 lim (\frac{y ^{2} + 3}{y - 1})

\int (8 e^{2} x) - (3 e^{- x}) d x

\frac{d}{d x} ((2 - 2 x)^{2} (3 x + 1)^{2})

\frac{d}{d x} (\frac{a + b x}{c + d x})

What is the integral of (5sqrt(x^2-1))/(x^2) ?
The integral of (5sqrt(x^2-1))/(x^2) is 5(-(sqrt(x^2-1))/x+ln|sqrt(x^2-1)+x|)+C