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

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

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integral of-3xsqrt(1-x^2)

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

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

Solution steps

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

Apply u-substitution

: \int - u^{2} d u

= - 3 \cdot \int - u^{2} d u

Take the constant out:

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

= - 3 (- \int u^{2} d u)

Apply the Power Rule

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

= - 3 (- \frac{u ^{3}}{3})

Substitute back

u = 1 - x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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