integral from 3 to 5 of sin(ln(x))

Lösung

\int_{3}^{5} sin (ln (x)) d x

\frac{5 sin ( ln ( 5 ) ) - 5 cos ( ln ( 5 ) ) - 3 sin ( ln ( 3 ) ) + 3 cos ( ln ( 3 ) )}{2}

Dezimale

1.94109 \dots

Schritte zur Lösung

\int_{3}^{5} sin (ln (x)) d x

Wende die partielle Integration an

= [x sin (ln (x)) - \int cos (ln (x)) d x]_{3}^{5}

\int cos (ln (x)) d x = \frac{1}{2} x sin (ln (x)) + \frac{1}{2} x cos (ln (x))

= [x sin (ln (x)) - (\frac{1}{2} x sin (ln (x)) + \frac{1}{2} x cos (ln (x)))]_{3}^{5}

Vereinfache

[x sin (ln (x)) - (\frac{1}{2} x sin (ln (x)) + \frac{1}{2} x cos (ln (x)))]_{3}^{5} : [\frac{1}{2} (x sin (ln (x)) - x cos (ln (x)))]_{3}^{5}

= [\frac{1}{2} (x sin (ln (x)) - x cos (ln (x)))]_{3}^{5}

Berechne die Grenzen:

\frac{5 sin ( ln ( 5 ) ) - 5 cos ( ln ( 5 ) ) - 3 sin ( ln ( 3 ) ) + 3 cos ( ln ( 3 ) )}{2}

= \frac{5 sin ( ln ( 5 ) ) - 5 cos ( ln ( 5 ) ) - 3 sin ( ln ( 3 ) ) + 3 cos ( ln ( 3 ) )}{2}