What is the integral of xsqrt(25-x^2) ?

The integral of xsqrt(25-x^2) is -1/3 (25-x^2)^{3/2}+C

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integral of xsqrt(25-x^2)

\int x 25 - x^{2} d x

- \frac{1}{3} (25 - x^{2})^{\frac{3}{2}} + C

Solution steps

\int x 25 - 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 = 25 - x^{2}

= - \frac{1}{2} \cdot \frac{2}{3} (25 - x^{2})^{\frac{3}{2}}

Simplify

- \frac{1}{2} \cdot \frac{2}{3} (25 - x^{2})^{\frac{3}{2}} : - \frac{1}{3} (25 - x^{2})^{\frac{3}{2}}

= - \frac{1}{3} (25 - x^{2})^{\frac{3}{2}}

Add a constant to the solution

= - \frac{1}{3} (25 - x^{2})^{\frac{3}{2}} + C

\int tan^{3} (d) x d x

\frac{d}{d θ} (r (θ) cos (θ))

3 x + y - 2 + y^{^{'}} (x - 1) = 0

\int_{1}^{2} π (2 - \frac{1}{2} x)^{2} d x

x = f x x^{3} - 4 x, (- 2, 0)

What is the integral of xsqrt(25-x^2) ?
The integral of xsqrt(25-x^2) is -1/3 (25-x^2)^{3/2}+C