What is the integral of sqrt(10-x^2)(5x) ?

The integral of sqrt(10-x^2)(5x) is -5/3 (10-x^2)^{3/2}+C

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integral of sqrt(10-x^2)(5x)

\int 10 - x^{2} (5 x) d x

- \frac{5}{3} (10 - x^{2})^{\frac{3}{2}} + C

Solution steps

\int 10 - x^{2} \cdot 5 x d x

Take the constant out:

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

= 5 \cdot \int 10 - x^{2} x d x

Apply u-substitution

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

= 5 \cdot \int - \frac{u}{2} d u

Take the constant out:

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

= 5 (- \frac{1}{2} \cdot \int u d u)

Apply the Power Rule

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

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

Substitute back

u = 10 - x^{2}

= 5 (- \frac{1}{2} \cdot \frac{2}{3} (10 - x^{2})^{\frac{3}{2}})

Simplify

5 (- \frac{1}{2} \cdot \frac{2}{3} (10 - x^{2})^{\frac{3}{2}}) : - \frac{5}{3} (10 - x^{2})^{\frac{3}{2}}

= - \frac{5}{3} (10 - x^{2})^{\frac{3}{2}}

Add a constant to the solution

= - \frac{5}{3} (10 - x^{2})^{\frac{3}{2}} + C

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What is the integral of sqrt(10-x^2)(5x) ?
The integral of sqrt(10-x^2)(5x) is -5/3 (10-x^2)^{3/2}+C