What is the integral from 0 to 2pi of-cos^2(x) ?

The integral from 0 to 2pi of-cos^2(x) is -pi

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integral from 0 to 2pi of-cos^2(x)

\int_{0}^{2 π} - cos^{2} (x) d x

- π

Decimal

- 3.14159 \dots

Solution steps

\int_{0}^{2 π} - cos^{2} (x) d x

Take the constant out:

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

= - \int_{0}^{2 π} cos^{2} (x) d x

Rewrite using trig identities

= - \int_{0}^{2 π} \frac{1 + cos ( 2 x )}{2} d x

Take the constant out:

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

= - \frac{1}{2} \cdot \int_{0}^{2 π} 1 + cos (2 x) 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

= - \frac{1}{2} (\int_{0}^{2 π} 1 d x + \int_{0}^{2 π} cos (2 x) d x)

\int_{0}^{2 π} 1 d x = 2 π

\int_{0}^{2 π} cos (2 x) d x = 0

= - \frac{1}{2} (2 π + 0)

Simplify

- \frac{1}{2} (2 π + 0) : - π

= - π

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What is the integral from 0 to 2pi of-cos^2(x) ?
The integral from 0 to 2pi of-cos^2(x) is -pi