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

The integral of (2x^3)/(sqrt(x^2-1)) is 2(1/3 (x^2-1)^{3/2}+sqrt(x^2-1))+C

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

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

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

Solution steps

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

Apply u-substitution

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

\int u^{2} d u = \frac{u ^{3}}{3}

\int 1 d u = u

= 2 (\frac{u ^{3}}{3} + u)

Substitute back

u = x^{2} - 1

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

Simplify

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

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

Add a constant to the solution

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

What is the integral of (2x^3)/(sqrt(x^2-1)) ?
The integral of (2x^3)/(sqrt(x^2-1)) is 2(1/3 (x^2-1)^{3/2}+sqrt(x^2-1))+C