integral from 0 to pi of sin^6(x/2)

Lösung

\int_{0}^{π} sin^{6} (\frac{x}{2}) d x

\frac{5 π}{16}

Dezimale

0.98174 \dots

Schritte zur Lösung

\int_{0}^{π} sin^{6} (\frac{x}{2}) d x

Wende U-Substitution an

= \int_{0}^{\frac{π}{2}} sin^{6} (u) \cdot 2 d u

Entferne die Konstante:

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

= 2 \cdot \int_{0}^{\frac{π}{2}} sin^{6} (u) d u

Wende integrale Reduktion an

= 2 (- [\frac{cos ( u ) sin ^{5} ( u )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \int_{0}^{\frac{π}{2}} sin^{4} (u) d u)

\int_{0}^{\frac{π}{2}} sin^{4} (u) d u = \frac{3 π}{16}

= 2 (- [\frac{cos ( u ) sin ^{5} ( u )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{16})

Vereinfache

2 (- [\frac{cos ( u ) sin ^{5} ( u )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{16}) : 2 (- [\frac{1}{6} sin^{5} (u) cos (u)]_{0}^{\frac{π}{2}} + \frac{5 π}{32})

= 2 (- [\frac{1}{6} sin^{5} (u) cos (u)]_{0}^{\frac{π}{2}} + \frac{5 π}{32})

Berechne die Grenzen:

0

= 2 (- 0 + \frac{5 π}{32})

Vereinfache

= \frac{5 π}{16}

\int_{0}^{a} sin^{2} (k x) d x

\int_{1}^{2} 2 x^{2} + x d x

\int_{2}^{4} (x^{3} - 2) - ln (x + 1) d x

\int_{- 2}^{- 1} \frac{1}{x ^{3}} + \frac{x ^{3}}{4} d x

\int_{3}^{4} (2 x - 5)^{4} d x