Question

Integrate cos(2theta)d(theta) from 0 to pi/4

Answer 1

\[\int\limits_{0}^{\Pi/4}\cos(2 \theta) d \theta\]

Answer 2

\[
\int\cos(2\theta)d\theta =\frac{\sin(2\theta)}{2}\\
\int\limits_{0}^{\Pi/4}\cos(2 \theta) d \theta=\frac{\sin(2\cdot\frac{\pi}{4})}{2}-\frac{\sin(2\cdot 0)}{2}
\]

Answer 3

why is it divided by 2? :o

Answer 4

Chain rule. If you derive \(\dfrac{\sin(2\theta)}{2}\) you'll get back \(\cos(2\theta)\). Without the division by 2 you would get back \(2\cos(2\theta)\).

Answer 5

ooohhhhh okay ^.^
what about

cos^2theta dtheta? is it cos^3theta/3?

Answer 6

\[
\int\cos^2x\ dx=\int\frac{1+\cos2x}{2}dx=\frac{x}{2}+\frac{\sin2x}4
\]
It isn't \(\cos^3x/3\) because if you derive that, you get \(\sin x\cdot\cos^2x\) because of the chain rule.

Answer 7

is that an identity? cos^2x = 1+cos2x/2? :o

Answer 8

Yep, it's an identity. You can derive it by combining the following two identities:\[
\sin^2x+\cos^2x=1\\\cos2x=\cos^2x-\sin^2x
\]

Answer 9

siiiick, thank you!!