What is the integral of (xsqrt(x^2-1)) ?

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

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integral of (xsqrt(x^2-1))

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

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

Solution steps

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

Apply u-substitution

: \int u^{2} d u

= \int u^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

= \frac{u ^{3}}{3}

Substitute back

u = x^{2} - 1

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

Simplify

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

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

Add a constant to the solution

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

x \to 1 - lim (4 x^{3} - e^{x - 1})

\int cos^{10} (x) sin (x) d x

\int sin (x) + 2 d x

f (x) = 3 e^{x} cos (x)

(x - 1)^{2} sin (\frac{1}{x - 1})

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