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

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

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integral of xsqrt(-x^2+1)

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

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

Solution steps

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

Apply u-substitution

= \int - u^{2} d u

Take the constant out:

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

= - \int u^{2} d u

Apply the Power Rule

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

Substitute back

u = - x^{2} + 1

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

Simplify

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

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

Add a constant to the solution

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

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