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

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

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integral of xsqrt(3-x^2)

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

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

Solution steps

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

Apply u-substitution

: \int - \frac{u}{2} d u

= \int - \frac{u}{2} d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int u d u

Apply the Power Rule

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

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

Substitute back

u = 3 - x^{2}

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

Simplify

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

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

Add a constant to the solution

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

x \to \infty lim (π^{\infty})

\int (x + 3)^{2} d x

2 x^{3} + 3 x^{2} - 12 x + 6

\frac{d}{d x} ((5 x) (5 x + 1))

\int e^{k} d k

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