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Beliebt Rechnen >

integral of arcsec(mx)

Lösung

\int arcsec (m x) d x

Lösung

x arcsec (m x) - \frac{1}{m} ln m x + m^{2} x^{2} - 1 + C

Schritte zur Lösung

\int arcsec (m x) d x

Wende U-Substitution an

= \int \frac{arcsec ( u )}{m} d u

Entferne die Konstante:

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

= \frac{1}{m} \cdot \int arcsec (u) d u

Wende die partielle Integration an

= \frac{1}{m} (u arcsec (u) - \int \frac{u}{u ^{2} u ^{2} - 1} d u)

\int \frac{u}{u ^{2} u ^{2} - 1} d u = ln u + u^{2} - 1

= \frac{1}{m} (u arcsec (u) - ln u + u^{2} - 1)

Setze in

u = m x

ein

= \frac{1}{m} (m x arcsec (m x) - ln m x + (m x)^{2} - 1)

Vereinfache

\frac{1}{m} (m x arcsec (m x) - ln m x + (m x)^{2} - 1) : x arcsec (m x) - \frac{1}{m} ln m x + m^{2} x^{2} - 1

= x arcsec (m x) - \frac{1}{m} ln m x + m^{2} x^{2} - 1

Füge eine Konstante zur Lösung hinzu

= x arcsec (m x) - \frac{1}{m} ln m x + m^{2} x^{2} - 1 + C

Graph

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