integral from 0 to x of 6/(1+6x^2)

Lösung

\int_{0}^{x} \frac{6}{1 + 6 x ^{2}} d x

6 arctan (6 x)

Schritte zur Lösung

\int_{0}^{x} \frac{6}{1 + 6 x ^{2}} d x

Entferne die Konstante:

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

= 6 \cdot \int_{0}^{x} \frac{1}{1 + 6 x ^{2}} d x

Wende integrale Substitution an

= 6 \cdot \int_{0}^{6 x} \frac{1}{6 ( 1 + u ^{2} )} d u

Entferne die Konstante:

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

= 6 \cdot \frac{1}{6} \cdot \int_{0}^{6 x} \frac{1}{1 + u ^{2}} d u

Nutze das gemeinsame Integral :

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

= 6 \cdot \frac{1}{6} [arctan (u)]_{0}^{6 x}

Vereinfache

6 \cdot \frac{1}{6} [arctan (u)]_{0}^{6 x} : 6 [arctan (u)]_{0}^{6 x}

= 6 [arctan (u)]_{0}^{6 x}

Berechne die Grenzen:

arctan (6 x)

= 6 arctan (6 x)

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