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Beliebt Rechnen >

integral from 0 to 3 of sqrt(x/(6-x))

Lösung

\int_{0}^{3} \frac{x}{6 - x} d x

Lösung

6 (\frac{π}{4} - \frac{1}{2})

+1

Dezimale

1.71238 \dots

Schritte zur Lösung

\int_{0}^{3} \frac{x}{6 - x} d x

Wende U-Substitution an

= \int_{6}^{3} - \frac{- u + 6}{u} d u

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

= - \int_{3}^{6} - \frac{- u + 6}{u} d u

Entferne die Konstante:

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

= - (- \int_{3}^{6} \frac{- u + 6}{u} d u)

Multipliziere aus

\frac{- u + 6}{u} : - 1 + \frac{6}{u}

= - (- \int_{3}^{6} - 1 + \frac{6}{u} d u)

Wende U-Substitution an

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

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

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

= - (- (- (- 6 [- \frac{v - 1}{v} - \int - \frac{1}{2 v v - 1} d v]_{1}^{2})))

\int - \frac{1}{2 v v - 1} d v = - arctan (v - 1)

= - (- (- (- 6 [- \frac{v - 1}{v} - (- arctan (v - 1))]_{1}^{2})))

Vereinfache

= 6 [- \frac{v - 1}{v} + arctan (v - 1)]_{1}^{2}

Berechne die Grenzen:

\frac{π}{4} - \frac{1}{2}

= 6 (\frac{π}{4} - \frac{1}{2})

Graph

Beliebte Beispiele

integral from 0 to 1 of 1/((3-2x)^2)

\int_{0}^{1} \frac{1}{( 3 - 2 x ) ^{2}} d x

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\int_{1}^{2} (x^{2})^{2} d x

integral from 0 to 1 of (3x)

\int_{0}^{1} (3 x) d x

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\int_{1}^{5} (9 x)^{3} d x

integral from-1 to 1 of 5/(3+x^2)

\int_{- 1}^{1} \frac{5}{3 + x ^{2}} d x