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

The integral of xsqrt(3x^2-1) is 1/9 (3x^2-1)^{3/2}+C

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integral of xsqrt(3x^2-1)

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

\frac{1}{9} (3 x^{2} - 1)^{\frac{3}{2}} + C

Solution steps

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

Apply u-substitution

: \int \frac{u}{6} d u

= \int \frac{u}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{1}{6} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 3 x^{2} - 1

= \frac{1}{6} \cdot \frac{2}{3} (3 x^{2} - 1)^{\frac{3}{2}}

Simplify

\frac{1}{6} \cdot \frac{2}{3} (3 x^{2} - 1)^{\frac{3}{2}} : \frac{1}{9} (3 x^{2} - 1)^{\frac{3}{2}}

= \frac{1}{9} (3 x^{2} - 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{9} (3 x^{2} - 1)^{\frac{3}{2}} + C

\frac{1}{( s ^{2} - 9 )}

\frac{d y}{d x} + 2 x^{2} y = 11 x^{2}

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

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

What is the integral of xsqrt(3x^2-1) ?
The integral of xsqrt(3x^2-1) is 1/9 (3x^2-1)^{3/2}+C