What is the integral from 0 to 2pi of (3+pi|sin(θ)|) ?

The integral from 0 to 2pi of (3+pi|sin(θ)|) is 10pi

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integral from 0 to 2pi of (3+pi|sin(θ)|)

\int_{0}^{2 π} (3 + π ∣ sin (θ) ∣) d θ

10 π

Decimal

31.41592 \dots

Solution steps

\int_{0}^{2 π} 3 + π ∣ sin (θ) ∣ d θ

Eliminate Absolutes

= \int_{0}^{π} 3 + π sin (θ) d θ + \int_{π}^{2 π} 3 + π (- sin (θ)) d θ

\int_{0}^{π} 3 + π sin (θ) d θ = 5 π

\int_{π}^{2 π} 3 + π (- sin (θ)) d θ = 5 π

= 5 π + 5 π

Simplify

= 10 π

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What is the integral from 0 to 2pi of (3+pi|sin(θ)|) ?
The integral from 0 to 2pi of (3+pi|sin(θ)|) is 10pi