integral from 0 to x^2 of sin(2sqrt(t))

Lösung

\int_{0}^{x^{2}} sin (2 t) d t

\frac{1}{2} sin (2 x) - x cos (2 x)

Schritte zur Lösung

\int_{0}^{x^{2}} sin (2 t) d t

Wende U-Substitution an

= \int_{0}^{x^{2}} sin (2 u) \cdot 2 u d u

Entferne die Konstante:

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

= 2 \cdot \int_{0}^{x^{2}} sin (2 u) u d u

Wende die partielle Integration an

= 2 [- \frac{1}{2} u cos (2 u) - \int - \frac{1}{2} cos (2 u) d u]_{0}^{x^{2}}

\int - \frac{1}{2} cos (2 u) d u = - \frac{1}{4} sin (2 u)

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

Vereinfache

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

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

Berechne die Grenzen:

\frac{1}{4} sin (2 x) - \frac{1}{2} x cos (2 x)

= 2 (\frac{1}{4} sin (2 x) - \frac{1}{2} x cos (2 x))

Vereinfache

= \frac{1}{2} sin (2 x) - x cos (2 x)

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