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

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

\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

Wende die Potenzregel an

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

Verwende die negative Winkelidentität:

cos (- x) = cos (x)

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

Berechne die Grenzen:

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

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

Vereinfache

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

\int_{\frac{π}{3}}^{\frac{π}{2}} cos^{3} (x) d x

\int_{- 2}^{2} e^{c o s (x)} sin (x) d x

\int_{- \frac{π}{4}}^{- \frac{π}{6}} 2 cos (x) d x

\int_{- \infty}^{x} f (t) d t

\int_{- 2}^{3} \frac{1}{x ^{2}} d x