integral from 0 to 5 of (3x)/((x-6)^2)

Lösung

\int_{0}^{5} \frac{3 x}{( x - 6 ) ^{2}} d x

3 (- ln (6) + 5)

Dezimale

9.62472 \dots

Schritte zur Lösung

\int_{0}^{5} \frac{3 x}{( x - 6 ) ^{2}} d x

Entferne die Konstante:

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

= 3 \cdot \int_{0}^{5} \frac{x}{( x - 6 ) ^{2}} d x

Wende U-Substitution an

= 3 \cdot \int_{- 6}^{- 1} \frac{u + 6}{u ^{2}} d u

Multipliziere aus

\frac{u + 6}{u ^{2}} : \frac{1}{u} + \frac{6}{u ^{2}}

= 3 \cdot \int_{- 6}^{- 1} \frac{1}{u} + \frac{6}{u ^{2}} 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

= 3 (\int_{- 6}^{- 1} \frac{1}{u} d u + \int_{- 6}^{- 1} \frac{6}{u ^{2}} d u)

\int_{- 6}^{- 1} \frac{1}{u} d u = - ln (6)

\int_{- 6}^{- 1} \frac{6}{u ^{2}} d u = 5

= 3 (- ln (6) + 5)

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

\frac{d}{d x} (sin (x^{6}))

\frac{d}{d x} (- \frac{24 x ( x ^{2} + 3 )}{( x ^{2} - 3 ) ^{4}})

\int cos (x) (6 + 8 sin^{2} (x)) d x

a re a 2 x^{2}, 8 x^{2}, - 3 x + 5