What is the integral from 0 to pi of cos(φ)sin(φ) ?

The integral from 0 to pi of cos(φ)sin(φ) is 0

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integral from 0 to pi of cos(φ)sin(φ)

\int_{0}^{π} cos (φ) sin (φ) d φ

0

Solution steps

\int_{0}^{π} cos (φ) sin (φ) d φ

Rewrite using trig identities

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

Take the constant out:

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

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

Apply u-substitution

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

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

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

Use the common integral:

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

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

Simplify

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

= \frac{1}{4} [- cos (u)]_{0}^{2 π}

Compute the boundaries:

0

= \frac{1}{4} \cdot 0

Simplify

= 0

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\int_{0}^{2 x} (y e^{x^{3}}) d y

What is the integral from 0 to pi of cos(φ)sin(φ) ?
The integral from 0 to pi of cos(φ)sin(φ) is 0