What is the integral from-2 to 2 of x^3cos(x/2)+1/2 sqrt(4-x^2) ?

The integral from-2 to 2 of x^3cos(x/2)+1/2 sqrt(4-x^2) is pi

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integral from-2 to 2 of x^3cos(x/2)+1/2 sqrt(4-x^2)

\int_{- 2}^{2} x^{3} cos (\frac{x}{2}) + \frac{1}{2} 4 - x^{2} d x

π

Decimal

3.14159 \dots

Solution steps

\int_{- 2}^{2} x^{3} cos (\frac{x}{2}) + \frac{1}{2} 4 - 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_{- 2}^{2} x^{3} cos (\frac{x}{2}) d x + \int_{- 2}^{2} \frac{1}{2} 4 - x^{2} d x

\int_{- 2}^{2} x^{3} cos (\frac{x}{2}) d x = 0

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

= 0 + π

Simplify

= π

\int_{0}^{\frac{π}{2}} 2 cos (t) d t

\int_{\frac{1}{2}}^{x} \frac{2}{t ^{2}} d t

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

\int_{0}^{3} x e^{- 0.4 x} d x

\int_{0}^{3} \frac{x}{16 - x ^{2}} d x

What is the integral from-2 to 2 of x^3cos(x/2)+1/2 sqrt(4-x^2) ?
The integral from-2 to 2 of x^3cos(x/2)+1/2 sqrt(4-x^2) is pi