integral from 1 to 3 of 2/(4+(x-1)^2)

Lösung

\int_{1}^{3} \frac{2}{4 + ( x - 1 ) ^{2}} d x

\frac{π}{4}

Dezimale

0.78539 \dots

Schritte zur Lösung

\int_{1}^{3} \frac{2}{4 + ( x - 1 ) ^{2}} d x

Entferne die Konstante:

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

= 2 \cdot \int_{1}^{3} \frac{1}{4 + ( x - 1 ) ^{2}} d x

Wende U-Substitution an

= 2 \cdot \int_{0}^{2} \frac{1}{4 + u ^{2}} d u

Wende integrale Substitution an

= 2 \cdot \int_{0}^{1} \frac{1}{2 ( v ^{2} + 1 )} d v

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

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

= 2 \cdot \frac{1}{2} [arctan (v)]_{0}^{1}

Vereinfache

2 \cdot \frac{1}{2} [arctan (v)]_{0}^{1} : [arctan (v)]_{0}^{1}

= [arctan (v)]_{0}^{1}

Berechne die Grenzen:

\frac{π}{4}

= \frac{π}{4}

y^{^{'}} = \frac{2 y}{x}

x \to 0 lim (\frac{x ^{2}}{x})

\frac{d}{d x} (2 sec (cos (x)) + 2 tan (2 x))

x \to 5 lim (\frac{x ^{2} + k x - 20}{x - 5})

s l o p e in t erce pt (1, 128), (4, 161)