What is the integral from-a to a of cos^2(t) ?

The integral from-a to a of cos^2(t) is a+1/2 sin(2a)

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integral from-a to a of cos^2(t)

\int_{- a}^{a} cos^{2} (t) d t

a + \frac{1}{2} sin (2 a)

Solution steps

\int_{- a}^{a} cos^{2} (t) d t

Rewrite using trig identities

= \int_{- a}^{a} \frac{1 + cos ( 2 t )}{2} d t

Take the constant out:

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

= \frac{1}{2} \cdot \int_{- a}^{a} 1 + cos (2 t) d t

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_{- a}^{a} 1 d t + \int_{- a}^{a} cos (2 t) d t)

\int_{- a}^{a} 1 d t = 2 a

\int_{- a}^{a} cos (2 t) d t = sin (2 a)

= \frac{1}{2} (2 a + sin (2 a))

Simplify

\frac{1}{2} (2 a + sin (2 a)) : a + \frac{1}{2} sin (2 a)

= a + \frac{1}{2} sin (2 a)

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What is the integral from-a to a of cos^2(t) ?
The integral from-a to a of cos^2(t) is a+1/2 sin(2a)