integral from-1/2 to 1/2 of 2/(1+4x^2)

Lösung

\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{2}{1 + 4 x ^{2}} d x

\frac{π}{2}

Dezimale

1.57079 \dots

Schritte zur Lösung

\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{2}{1 + 4 x ^{2}} d x

Entferne die Konstante:

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

= 2 \cdot \int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{1}{1 + 4 x ^{2}} d x

Wende integrale Substitution an

= 2 \cdot \int_{- 1}^{1} \frac{1}{2 ( 1 + u ^{2} )} d u

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

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

= 2 \cdot \frac{1}{2} [arctan (u)]_{- 1}^{1}

Vereinfache

2 \cdot \frac{1}{2} [arctan (u)]_{- 1}^{1} : [arctan (u)]_{- 1}^{1}

= [arctan (u)]_{- 1}^{1}

Berechne die Grenzen:

\frac{π}{2}

= \frac{π}{2}

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