What is the integral of (2(x^2-1)^{1/2})/x ?

The integral of (2(x^2-1)^{1/2})/x is 2(-arctan((x^2-1)^{1/2})+(x^2-1)^{1/2})+C

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integral of (2(x^2-1)^{1/2})/x

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{u ^{2}}{u ^{2} + 1} d u

= 2 \cdot \int \frac{u ^{2}}{u ^{2} + 1} d u

\frac{u ^{2}}{u ^{2} + 1} = - \frac{1}{u ^{2} + 1} + 1

= 2 \cdot \int - \frac{1}{u ^{2} + 1} + 1 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (- \int \frac{1}{u ^{2} + 1} d u + \int 1 d u)

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

\int 1 d u = u

= 2 (- arctan (u) + u)

Substitute back

u = (x^{2} - 1)^{\frac{1}{2}}

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

Add a constant to the solution

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

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What is the integral of (2(x^2-1)^{1/2})/x ?
The integral of (2(x^2-1)^{1/2})/x is 2(-arctan((x^2-1)^{1/2})+(x^2-1)^{1/2})+C