integral from 0 to 1 of x/(x^2+3x+2)

Lösung

\int_{0}^{1} \frac{x}{x ^{2} + 3 x + 2} d x

- 3 ln (2) + 2 ln (3)

Dezimale

0.11778 \dots

Schritte zur Lösung

\int_{0}^{1} \frac{x}{x ^{2} + 3 x + 2} d x

Ermittle den Partialbruch von

\frac{x}{x ^{2} + 3 x + 2} : - \frac{1}{x + 1} + \frac{2}{x + 2}

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

Wende die Summenregel an:

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

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

\int_{0}^{1} \frac{1}{x + 1} d x = ln (2)

\int_{0}^{1} \frac{2}{x + 2} d x = 2 (ln (3) - ln (2))

= - ln (2) + 2 (ln (3) - ln (2))

Vereinfache

- ln (2) + 2 (ln (3) - ln (2)) : - 3 ln (2) + 2 ln (3)

= - 3 ln (2) + 2 ln (3)

\int_{0}^{1} x^{3} - x d x

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

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\int_{1}^{7} \frac{1}{x ^{2}} d x