What is the integral of 3xsqrt(5-2x^2) ?

The integral of 3xsqrt(5-2x^2) is -1/2 (5-2x^2)^{3/2}+C

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integral of 3xsqrt(5-2x^2)

\int 3 x 5 - 2 x^{2} d x

- \frac{1}{2} (5 - 2 x^{2})^{\frac{3}{2}} + C

Solution steps

\int 3 x 5 - 2 x^{2} d x

Take the constant out:

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

= 3 \cdot \int x 5 - 2 x^{2} d x

Apply u-substitution

: \int - \frac{u}{4} d u

= 3 \cdot \int - \frac{u}{4} d u

Take the constant out:

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

= 3 (- \frac{1}{4} \cdot \int u d u)

Apply the Power Rule

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

= 3 (- \frac{1}{4} \cdot \frac{2}{3} u^{\frac{3}{2}})

Substitute back

u = 5 - 2 x^{2}

= 3 (- \frac{1}{4} \cdot \frac{2}{3} (5 - 2 x^{2})^{\frac{3}{2}})

Simplify

3 (- \frac{1}{4} \cdot \frac{2}{3} (5 - 2 x^{2})^{\frac{3}{2}}) : - \frac{1}{2} (5 - 2 x^{2})^{\frac{3}{2}}

= - \frac{1}{2} (5 - 2 x^{2})^{\frac{3}{2}}

Add a constant to the solution

= - \frac{1}{2} (5 - 2 x^{2})^{\frac{3}{2}} + C

\int cos (y^{2}) d y

\frac{e ^{- 2 s}}{s ( s + 1 )}

x \cdot \frac{d y}{d x} = - y + 2 x^{2} y

\frac{\partial}{\partial y} (x^{2} + 2 y^{2} - 1)

\int \frac{9 x ^{8} + 8}{x ^{9} + 8 x} d x

What is the integral of 3xsqrt(5-2x^2) ?
The integral of 3xsqrt(5-2x^2) is -1/2 (5-2x^2)^{3/2}+C