What is the integral from-w/2 to w of cos^3(x)sin(x) ?

The integral from-w/2 to w of cos^3(x)sin(x) is -(cos^4(w)-cos^4(w/2))/4

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integral from-w/2 to w of cos^3(x)sin(x)

\int_{- \frac{w}{2}}^{w} cos^{3} (x) sin (x) d x

- \frac{cos ^{4} ( w ) - cos ^{4} ( \frac{w}{2} )}{4}

Solution steps

\int_{- \frac{w}{2}}^{w} cos^{3} (x) sin (x) d x

Apply u-substitution

= \int_{c o s (- \frac{w}{2})}^{c o s (w)} - u^{3} d u

Take the constant out:

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

= - \int_{c o s (- \frac{w}{2})}^{c o s (w)} u^{3} d u

Apply the Power Rule

= - [\frac{u ^{4}}{4}]_{c o s (- \frac{w}{2})}^{c o s (w)}

Use the negative angle identity:

cos (- x) = cos (x)

= - [\frac{u ^{4}}{4}]_{c o s (\frac{w}{2})}^{c o s (w)}

Compute the boundaries:

\frac{cos ^{4} ( w )}{4} - \frac{cos ^{4} ( \frac{w}{2} )}{4}

= - (\frac{cos ^{4} ( w )}{4} - \frac{cos ^{4} ( \frac{w}{2} )}{4})

Simplify

= - \frac{cos ^{4} ( w ) - cos ^{4} ( \frac{w}{2} )}{4}

What is the integral from-w/2 to w of cos^3(x)sin(x) ?
The integral from-w/2 to w of cos^3(x)sin(x) is -(cos^4(w)-cos^4(w/2))/4