English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from 0 to pi of (cos(x))^6

Lösung

\int_{0}^{π} (cos (x))^{6} d x

\frac{5 π}{16}

Dezimale

0.98174 \dots

Schritte zur Lösung

\int_{0}^{π} (cos (x))^{6} d x

Wende U-Substitution an

= \int_{1}^{- 1} - \frac{u ^{6}}{1 - u ^{2}} d u

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

= - \int_{- 1}^{1} - \frac{u ^{6}}{1 - u ^{2}} d u

Entferne die Konstante:

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

= - (- \int_{- 1}^{1} \frac{u ^{6}}{1 - u ^{2}} d u)

Trigonometrische Substitution anwenden

= - (- \int_{- \frac{π}{2}}^{\frac{π}{2}} sin^{6} (v) d v)

Wende integrale Reduktion an

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

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

Wende integrale Reduktion an

= - (- (- [\frac{cos ( v ) sin ^{5} ( v )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{cos ( v ) sin ^{3} ( v )}{4}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3}{4} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} sin^{2} (v) d v)))

= - (- (- [\frac{cos ( v ) sin ^{5} ( v )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{cos ( v ) sin ^{3} ( v )}{4}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3}{4} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} sin^{2} (v) d v)))

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= - (- (- [\frac{cos ( v ) sin ^{5} ( v )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{cos ( v ) sin ^{3} ( v )}{4}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3}{4} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 - cos ( 2 v )}{2} d v)))

Entferne die Konstante:

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

= - (- (- [\frac{cos ( v ) sin ^{5} ( v )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{cos ( v ) sin ^{3} ( v )}{4}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3}{4} \cdot \frac{1}{2} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 - cos (2 v) d v)))

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (- (- [\frac{cos ( v ) sin ^{5} ( v )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{cos ( v ) sin ^{3} ( v )}{4}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3}{4} \cdot \frac{1}{2} (\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d v - \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 v) d v))))

\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d v = π

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 v) d v = 0

= - (- (- [\frac{cos ( v ) sin ^{5} ( v )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{cos ( v ) sin ^{3} ( v )}{4}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3}{4} \cdot \frac{1}{2} (π - 0))))

Vereinfache

- (- (- [\frac{cos ( v ) sin ^{5} ( v )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{cos ( v ) sin ^{3} ( v )}{4}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3}{4} \cdot \frac{1}{2} (π - 0)))) : - [\frac{1}{6} sin^{5} (v) cos (v)]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{1}{4} sin^{3} (v) cos (v)]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3 π}{8})

= - [\frac{1}{6} sin^{5} (v) cos (v)]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{1}{4} sin^{3} (v) cos (v)]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3 π}{8})

Berechne die Grenzen:

0

Berechne die Grenzen:

0

= - 0 + \frac{5}{6} (- 0 + \frac{3 π}{8})

Vereinfache

= \frac{5 π}{16}

\int_{2}^{4} 2 cos (π x) d x

\int \frac{- 1}{1 - 2 x + x ^{2}} d x

x \to 0 lim (e^{c s c (x)})

d er i v a t i v e f (x) = \frac{4 x}{9 e ^{5 x}}

s l o p e in t erce pt (5, - 2), (- 6, 4)