integral from 0 to pi/2 of 1/4 cos^4(x)

Lösung

\int_{0}^{\frac{π}{2}} \frac{1}{4} cos^{4} (x) d x

\frac{3 π}{64}

Dezimale

0.14726 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{2}} \frac{1}{4} cos^{4} (x) d x

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int_{0}^{\frac{π}{2}} cos^{4} (x) d x

Vereinfache

cos^{4} (x) : cos^{3} (x) cos (x)

= \frac{1}{4} \cdot \int_{0}^{\frac{π}{2}} cos^{3} (x) cos (x) d x

Wende die partielle Integration an

= \frac{1}{4} [cos^{3} (x) sin (x) - \int - 3 cos^{2} (x) sin^{2} (x) d x]_{0}^{\frac{π}{2}}

\int - 3 cos^{2} (x) sin^{2} (x) d x = - \frac{3}{8} (x - \frac{1}{4} sin (4 x))

= \frac{1}{4} [cos^{3} (x) sin (x) - (- \frac{3}{8} (x - \frac{1}{4} sin (4 x)))]_{0}^{\frac{π}{2}}

Vereinfache

\frac{1}{4} [cos^{3} (x) sin (x) - (- \frac{3}{8} (x - \frac{1}{4} sin (4 x)))]_{0}^{\frac{π}{2}} : \frac{1}{4} [\frac{1}{32} (32 cos^{3} (x) sin (x) + 3 (4 x - sin (4 x)))]_{0}^{\frac{π}{2}}

= \frac{1}{4} [\frac{1}{32} (32 cos^{3} (x) sin (x) + 3 (4 x - sin (4 x)))]_{0}^{\frac{π}{2}}

Berechne die Grenzen:

\frac{3 π}{16}

= \frac{1}{4} \cdot \frac{3 π}{16}

Vereinfache

= \frac{3 π}{64}

\int_{- 1}^{3} x + 3 d x

\int_{0}^{1} \frac{1}{9 + x ^{2}} d x

\int_{0}^{π} ln (x) d x

\int_{0}^{x} x y d y

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