What is the integral from 0 to 1 of x+sqrt(1-x^2) ?

The integral from 0 to 1 of x+sqrt(1-x^2) is 1/2+pi/4

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integral from 0 to 1 of x+sqrt(1-x^2)

\int_{0}^{1} x + 1 - x^{2} d x

\frac{1}{2} + \frac{π}{4}

Decimal

1.28539 \dots

Solution steps

\int_{0}^{1} 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_{0}^{1} x d x + \int_{0}^{1} 1 - x^{2} d x

\int_{0}^{1} x d x = \frac{1}{2}

\int_{0}^{1} 1 - x^{2} d x = \frac{π}{4}

= \frac{1}{2} + \frac{π}{4}

What is the integral from 0 to 1 of x+sqrt(1-x^2) ?
The integral from 0 to 1 of x+sqrt(1-x^2) is 1/2+pi/4