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

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

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integral of x*(sqrt(x^2-3))

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

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

Solution steps

\int x x^{2} - 3 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} - 3

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

Simplify

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

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

Add a constant to the solution

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

y = x sin (\frac{1}{x})

3 x^{2} (x^{3} + 1)^{7}

x \to 0 lim (\frac{sin ^{2} ( 4 x )}{x})

x \to 0 + lim (- \frac{1}{sin ( 2 x )})

4 (t + 3)

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