What is the integral from-4pi to pi of cos^2(x) ?

The integral from-4pi to pi of cos^2(x) is (5pi)/2

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integral from-4pi to pi of cos^2(x)

\int_{- 4 π}^{π} cos^{2} (x) d x

\frac{5 π}{2}

Decimal

7.85398 \dots

Solution steps

\int_{- 4 π}^{π} cos^{2} (x) d x

Rewrite using trig identities

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

\int_{- 4 π}^{π} 1 d x = 5 π

\int_{- 4 π}^{π} cos (2 x) d x = 0

= \frac{1}{2} (5 π + 0)

Simplify

\frac{1}{2} (5 π + 0) : \frac{5 π}{2}

= \frac{5 π}{2}

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

\int_{0}^{π} - x cos (x) d x

\int_{- 2}^{1} (2 t^{2} - 1)^{2} d t

\int_{1}^{x} 1 + t d t

\int_{0}^{1} (x^{\frac{1}{2}} - x^{2}) d x

What is the integral from-4pi to pi of cos^2(x) ?
The integral from-4pi to pi of cos^2(x) is (5pi)/2