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

Lösung

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

\frac{π}{4 a}

Schritte zur Lösung

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

Wende integrale Substitution an

= \int_{0}^{1} \frac{1}{a ( u ^{2} + 1 )} d u

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

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

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

Berechne die Grenzen:

\frac{π}{4}

= \frac{1}{a} \cdot \frac{π}{4}

Vereinfache

= \frac{π}{4 a}

\int_{0}^{x} 9 + x^{2} d x

\int_{0}^{5} (2 x + 1) d x

\int_{0}^{2} (x^{2} - 9 x + 14) d x

\int_{0}^{4} \frac{1}{3 x - 1} d x

\int_{- 1}^{0} e^{x} cos (e^{x}) d x