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

4 x y^{^{'}} = 2 x e^{x} - 4 y + 6 x^{2}

\int x sin^{3} (x^{2}) cos (x^{2}) d x

\frac{d y}{d x} = x, y (11) = 5

y = 5 sec (x), at x = \frac{π}{3}

\frac{d}{d x} (- 5 e^{- 4 x})

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