What is the integral from pi/4 to pi/2 of 4sin(2θ) ?

The integral from pi/4 to pi/2 of 4sin(2θ) is 2

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integral from pi/4 to pi/2 of 4sin(2θ)

\int_{\frac{π}{4}}^{\frac{π}{2}} 4 sin (2 θ) d θ

2

Solution steps

\int_{\frac{π}{4}}^{\frac{π}{2}} 4 sin (2 θ) d θ

Take the constant out:

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

= 4 \cdot \int_{\frac{π}{4}}^{\frac{π}{2}} sin (2 θ) d θ

Apply u-substitution

: \int_{\frac{π}{2}}^{π} sin (u) \frac{1}{2} d u

= 4 \cdot \int_{\frac{π}{2}}^{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

= 4 \cdot \frac{1}{2} \cdot \int_{\frac{π}{2}}^{π} sin (u) d u

Use the common integral:

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

= 4 \cdot \frac{1}{2} [- cos (u)]_{\frac{π}{2}}^{π}

Simplify

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

= 2 [- cos (u)]_{\frac{π}{2}}^{π}

Compute the boundaries:

1

= 2 \cdot 1

Simplify

= 2

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What is the integral from pi/4 to pi/2 of 4sin(2θ) ?
The integral from pi/4 to pi/2 of 4sin(2θ) is 2