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

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

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

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

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

Solution steps

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int x^{2} d x = \frac{x ^{3}}{3}

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

Add a constant to the solution

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

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