integral from 0 to 1/3 of 5/(1+9x^2)

Lösung

\int_{0}^{\frac{1}{3}} \frac{5}{1 + 9 x ^{2}} d x

\frac{5 π}{12}

Dezimale

1.30899 \dots

Schritte zur Lösung

\int_{0}^{\frac{1}{3}} \frac{5}{1 + 9 x ^{2}} d x

Entferne die Konstante:

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

= 5 \cdot \int_{0}^{\frac{1}{3}} \frac{1}{1 + 9 x ^{2}} d x

Wende integrale Substitution an

= 5 \cdot \int_{0}^{1} \frac{1}{3 ( 1 + u ^{2} )} d u

Entferne die Konstante:

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

= 5 \cdot \frac{1}{3} \cdot \int_{0}^{1} \frac{1}{1 + u ^{2}} d u

Nutze das gemeinsame Integral :

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

= 5 \cdot \frac{1}{3} [arctan (u)]_{0}^{1}

Vereinfache

5 \cdot \frac{1}{3} [arctan (u)]_{0}^{1} : \frac{5}{3} [arctan (u)]_{0}^{1}

= \frac{5}{3} [arctan (u)]_{0}^{1}

Berechne die Grenzen:

\frac{π}{4}

= \frac{5}{3} \cdot \frac{π}{4}

Vereinfache

= \frac{5 π}{12}

\frac{d}{d x} (\frac{1}{2} ln (\frac{x}{2}))

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