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Beliebt Rechnen >

integral of (6x-5)/(x^2+100)

Lösung

\int \frac{6 x - 5}{x ^{2} + 100} d x

Lösung

\frac{1}{2} (6 ln \frac{x ^{2}}{100} + 1 - arctan (\frac{x}{10})) + C

Schritte zur Lösung

\int \frac{6 x - 5}{x ^{2} + 100} d x

Wende integrale Substitution an

= \int \frac{12 u - 1}{2 ( u ^{2} + 1 )} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{12 u - 1}{u ^{2} + 1} d u

Multipliziere aus

\frac{12 u - 1}{u ^{2} + 1} : \frac{12 u}{u ^{2} + 1} - \frac{1}{u ^{2} + 1}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int \frac{12 u}{u ^{2} + 1} d u - \int \frac{1}{u ^{2} + 1} d u)

\int \frac{12 u}{u ^{2} + 1} d u = 6 ln u^{2} + 1

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= \frac{1}{2} (6 ln u^{2} + 1 - arctan (u))

Setze in

u = \frac{x}{10}

ein

= \frac{1}{2} (6 ln (\frac{x}{10})^{2} + 1 - arctan (\frac{x}{10}))

(\frac{x}{10})^{2} = \frac{x ^{2}}{100}

= \frac{1}{2} (6 ln \frac{x ^{2}}{100} + 1 - arctan (\frac{x}{10}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (6 ln \frac{x ^{2}}{100} + 1 - arctan (\frac{x}{10})) + C

Graph

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