integral from 1 to 5 of ((x-1)^3)/(x^2)

Lösung

\int_{1}^{5} \frac{( x - 1 ) ^{3}}{x ^{2}} d x

3 ln (5) - \frac{4}{5}

Dezimale

4.02831 \dots

Schritte zur Lösung

\int_{1}^{5} \frac{( x - 1 ) ^{3}}{x ^{2}} d x

Multipliziere aus

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

= \int_{1}^{5} x - 3 + \frac{3}{x} - \frac{1}{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_{1}^{5} x d x - \int_{1}^{5} 3 d x + \int_{1}^{5} \frac{3}{x} d x - \int_{1}^{5} \frac{1}{x ^{2}} d x

\int_{1}^{5} x d x = 12

\int_{1}^{5} 3 d x = 12

\int_{1}^{5} \frac{3}{x} d x = 3 ln (5)

\int_{1}^{5} \frac{1}{x ^{2}} d x = \frac{4}{5}

= 12 - 12 + 3 ln (5) - \frac{4}{5}

Vereinfache

= 3 ln (5) - \frac{4}{5}

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

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