integral from 0 to 4 of 2/(1+sqrt(x))

Lösung

\int_{0}^{4} \frac{2}{1 + x} d x

4 (2 - ln (3))

Dezimale

3.60555 \dots

Schritte zur Lösung

\int_{0}^{4} \frac{2}{1 + x} d x

Entferne die Konstante:

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

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

Wende U-Substitution an

= 2 \cdot \int_{1}^{3} \frac{2 ( u - 1 )}{u} d u

Entferne die Konstante:

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

= 2 \cdot 2 \cdot \int_{1}^{3} \frac{u - 1}{u} d u

Multipliziere aus

\frac{u - 1}{u} : 1 - \frac{1}{u}

= 2 \cdot 2 \cdot \int_{1}^{3} 1 - \frac{1}{u} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 \cdot 2 (\int_{1}^{3} 1 d u - \int_{1}^{3} \frac{1}{u} d u)

\int_{1}^{3} 1 d u = 2

\int_{1}^{3} \frac{1}{u} d u = ln (3)

= 2 \cdot 2 (2 - ln (3))

Vereinfache

= 4 (2 - ln (3))

\int_{0}^{\frac{π}{4}} (9 sec^{2} (x)) d x

\int_{0}^{3} (y - 2) (2 y + 1) d y

\int_{0}^{10} ∣ x - 7 ∣ d x

\int_{1}^{e} x^{4} (ln (x))^{2} d x

\int_{0}^{4} x + 3 d x