What is the integral from 0 to pi/2 of 2sin(2θ) ?

The integral from 0 to pi/2 of 2sin(2θ) is 2

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integral from 0 to pi/2 of 2sin(2θ)

\int_{0}^{\frac{π}{2}} 2 sin (2 θ) d θ

2

Solution steps

\int_{0}^{\frac{π}{2}} 2 sin (2 θ) d θ

Take the constant out:

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

= 2 \cdot \int_{0}^{\frac{π}{2}} sin (2 θ) d θ

Apply u-substitution

= 2 \cdot \int_{0}^{π} sin (u) \frac{1}{2} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int_{0}^{π} sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= 2 \cdot \frac{1}{2} [- cos (u)]_{0}^{π}

Simplify

2 \cdot \frac{1}{2} [- cos (u)]_{0}^{π} : [- cos (u)]_{0}^{π}

= [- cos (u)]_{0}^{π}

Compute the boundaries:

2

= 2

What is the integral from 0 to pi/2 of 2sin(2θ) ?
The integral from 0 to pi/2 of 2sin(2θ) is 2