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

Lösung

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

4 π

Dezimale

12.56637 \dots

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

= \frac{1}{2} [16 x sin^{2} (x) - \int 32 x sin (x) cos (x) d x]_{0}^{π}

\int 32 x sin (x) cos (x) d x = 16 (- \frac{1}{2} x cos (2 x) + \frac{1}{4} sin (2 x))

= \frac{1}{2} [16 x sin^{2} (x) - 16 (- \frac{1}{2} x cos (2 x) + \frac{1}{4} sin (2 x))]_{0}^{π}

Berechne die Grenzen:

8 π

= \frac{1}{2} \cdot 8 π

Vereinfache

= 4 π

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