What is the integral from pi to 0 of sin(x)cos^3(x) ?

The integral from pi to 0 of sin(x)cos^3(x) is 0

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integral from pi to 0 of sin(x)cos^3(x)

\int_{π}^{0} sin (x) cos^{3} (x) d x

0

Solution steps

\int_{π}^{0} sin (x) cos^{3} (x) d x

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{0}^{π} sin (x) cos^{3} (x) d x

Apply u-substitution

: \int_{1}^{- 1} - u^{3} d u

= - \int_{1}^{- 1} - u^{3} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - (- \int_{- 1}^{1} - u^{3} d u)

Take the constant out:

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

= - (- (- \int_{- 1}^{1} u^{3} d u))

Apply the Power Rule

: [\frac{u ^{4}}{4}]_{- 1}^{1}

= - (- (- [\frac{u ^{4}}{4}]_{- 1}^{1}))

Simplify

= - [\frac{u ^{4}}{4}]_{- 1}^{1}

Compute the boundaries:

0

= - 0

Simplify

= 0

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What is the integral from pi to 0 of sin(x)cos^3(x) ?
The integral from pi to 0 of sin(x)cos^3(x) is 0