What is the integral of 1/2 (2cos(θ))^2 ?

The integral of 1/2 (2cos(θ))^2 is 2θcos^2(θ)-θcos(2θ)+1/2 sin(2θ)+C

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integral of 1/2 (2cos(θ))^2

\int \frac{1}{2} (2 cos (θ))^{2} d θ

2 θ cos^{2} (θ) - θ cos (2 θ) + \frac{1}{2} sin (2 θ) + C

Solution steps

\int \frac{1}{2} (2 cos (θ))^{2} d θ

Take the constant out:

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

= \frac{1}{2} \cdot \int (2 cos (θ))^{2} d θ

Apply Integration By Parts

: 4 θ cos^{2} (θ) - \int - 4 θ sin (2 θ) d θ

= \frac{1}{2} (4 θ cos^{2} (θ) - \int - 4 θ sin (2 θ) d θ)

\int - 4 θ sin (2 θ) d θ = 2 θ cos (2 θ) - sin (2 θ)

= \frac{1}{2} (4 θ cos^{2} (θ) - (2 θ cos (2 θ) - sin (2 θ)))

Simplify

\frac{1}{2} (4 θ cos^{2} (θ) - (2 θ cos (2 θ) - sin (2 θ))) : 2 θ cos^{2} (θ) - θ cos (2 θ) + \frac{1}{2} sin (2 θ)

= 2 θ cos^{2} (θ) - θ cos (2 θ) + \frac{1}{2} sin (2 θ)

Add a constant to the solution

= 2 θ cos^{2} (θ) - θ cos (2 θ) + \frac{1}{2} sin (2 θ) + C

\int 200 - x d x

\int_{0}^{6} 2 x - (x^{2} - 4 x) d x

x \to 0 + lim (x^{2})

\frac{\partial}{\partial x} (x^{2} - 10 y^{2})

x \to 0 lim (2^{x} - 5^{x})

What is the integral of 1/2 (2cos(θ))^2 ?
The integral of 1/2 (2cos(θ))^2 is 2θcos^2(θ)-θcos(2θ)+1/2 sin(2θ)+C