What is the integral from 0 to 2pi of 8sin(θ)cos(θ) ?

The integral from 0 to 2pi of 8sin(θ)cos(θ) is 0

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integral from 0 to 2pi of 8sin(θ)cos(θ)

\int_{0}^{2 π} 8 sin (θ) cos (θ) d θ

0

Solution steps

\int_{0}^{2 π} 8 sin (θ) cos (θ) d θ

Take the constant out:

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

= 8 \cdot \int_{0}^{2 π} sin (θ) cos (θ) d θ

Rewrite using trig identities

= 8 \cdot \int_{0}^{2 π} \frac{sin ( 2 θ )}{2} d θ

Take the constant out:

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

= 8 \cdot \frac{1}{2} \cdot \int_{0}^{2 π} sin (2 θ) d θ

Apply u-substitution

: \int_{0}^{4 π} sin (u) \frac{1}{2} d u

= 8 \cdot \frac{1}{2} \cdot \int_{0}^{4 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

= 8 \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \int_{0}^{4 π} sin (u) d u

Use the common integral:

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

= 8 \cdot \frac{1}{2} \cdot \frac{1}{2} [- cos (u)]_{0}^{4 π}

Simplify

8 \cdot \frac{1}{2} \cdot \frac{1}{2} [- cos (u)]_{0}^{4 π} : 2 [- cos (u)]_{0}^{4 π}

= 2 [- cos (u)]_{0}^{4 π}

Compute the boundaries:

0

= 2 \cdot 0

Simplify

= 0

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What is the integral from 0 to 2pi of 8sin(θ)cos(θ) ?
The integral from 0 to 2pi of 8sin(θ)cos(θ) is 0