integral from 0 to 1 of 5/(sqrt(x)(1+x))

Lösung

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

\frac{5 π}{2}

Dezimale

7.85398 \dots

Schritte zur Lösung

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

Entferne die Konstante:

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

= 5 \cdot \int_{0}^{1} \frac{1}{x ( 1 + x )} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

= 5 \cdot 2 \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 2 [arctan (u)]_{0}^{1}

Vereinfache

= 10 [arctan (u)]_{0}^{1}

Berechne die Grenzen:

\frac{π}{4}

= 10 \cdot \frac{π}{4}

Vereinfache

= \frac{5 π}{2}

\frac{d y}{d t} = k (80 - y) (50 - y)

x \to 4 lim (1 - 2 x)

\int_{0}^{1} t e^{2 t} d t

t an g e n t tan (x), at x = - \frac{π}{3}

\int_{0}^{14} 10 e^{x} d x