integral from 1 to 9 of 1/(8-sqrt(x))

Lösung

\int_{1}^{9} \frac{1}{8 - x} d x

2 (- 2 + 8 (ln (7) - ln (5)))

Dezimale

1.38355 \dots

Schritte zur Lösung

\int_{1}^{9} \frac{1}{8 - x} d x

Wende U-Substitution an

= \int_{1}^{3} \frac{2 u}{8 - u} d u

Entferne die Konstante:

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

= 2 \cdot \int_{1}^{3} \frac{u}{8 - u} d u

Wende U-Substitution an

= 2 \cdot \int_{7}^{5} - \frac{- v + 8}{v} d v

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 2 (- \int_{5}^{7} - \frac{- v + 8}{v} d v)

Entferne die Konstante:

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

= 2 (- (- \int_{5}^{7} \frac{- v + 8}{v} d v))

Multipliziere aus

\frac{- v + 8}{v} : - 1 + \frac{8}{v}

= 2 (- (- \int_{5}^{7} - 1 + \frac{8}{v} d v))

Wende die Summenregel an:

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

= 2 (- (- (- \int_{5}^{7} 1 d v + \int_{5}^{7} \frac{8}{v} d v)))

\int_{5}^{7} 1 d v = 2

\int_{5}^{7} \frac{8}{v} d v = 8 (ln (7) - ln (5))

= 2 (- (- (- 2 + 8 (ln (7) - ln (5)))))

Vereinfache

= 2 (- 2 + 8 (ln (7) - ln (5)))

\int \frac{x ^{3} - 2}{x + 1} d x

\int 2 x + 8 d x

\int 6 x^{2} e^{x} d x

\int \frac{u}{u - 1} d u

d er i v a t i v e f (x) = x cos (x) sin (x)