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Beliebt Rechnen >

integral from 0 to 1/2 of sqrt(1-4x^2)

Lösung

\int_{0}^{\frac{1}{2}} 1 - 4 x^{2} d x

Lösung

\frac{π}{8}

+1

Dezimale

0.39269 \dots

Schritte zur Lösung

\int_{0}^{\frac{1}{2}} 1 - 4 x^{2} d x

Trigonometrische Substitution anwenden

= \int_{0}^{\frac{π}{2}} \frac{1}{2} cos^{2} (u) d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} \frac{1 + cos ( 2 u )}{2} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} 1 + cos (2 u) d u

Wende die Summenregel an:

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

= \frac{1}{2} \cdot \frac{1}{2} (\int_{0}^{\frac{π}{2}} 1 d u + \int_{0}^{\frac{π}{2}} cos (2 u) d u)

\int_{0}^{\frac{π}{2}} 1 d u = \frac{π}{2}

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

= \frac{1}{2} \cdot \frac{1}{2} (\frac{π}{2} + 0)

Vereinfache

\frac{1}{2} \cdot \frac{1}{2} (\frac{π}{2} + 0) : \frac{π}{8}

= \frac{π}{8}

Graph

Beliebte Beispiele

integral of 6/(xsqrt(x^2+1))

\int \frac{6}{x x ^{2} + 1} d x

integral of (x+1)e^{9x^2+18x}

\int (x + 1) e^{9 x^{2} + 18 x} d x

(d^2y)/(dx^2)-4(dy)/(dx)+13y=8sin(3x),y(0)=1,y^'(0)=2

\frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} + 13 y = 8 sin (3 x), y (0) = 1, y^{^{'}} (0) = 2

limit as x approaches 1 of (9x^2)+30x+25

x \to 1 lim ((9 x^{2}) + 30 x + 25)

laplacetransform 4y

l a pl a ce t r an s f or m 4 y