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

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

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integral of x(x+sqrt(1-x^2))

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

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

Solution steps

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

Expand

x (x + 1 - x^{2}) : x^{2} + x 1 - x^{2}

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

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

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

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

Add a constant to the solution

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

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