integral from 0 to 1 of (x-x^2)sin(npix)

Lösung

\int_{0}^{1} (x - x^{2}) sin (nπ x) d x

- \frac{2 ( - 1 ) ^{n} - 2}{π ^{3} n ^{3}}

Schritte zur Lösung

\int_{0}^{1} (x - x^{2}) sin (nπ x) d x

Multipliziere aus

(x - x^{2}) sin (nπ x) : x sin (πn x) - x^{2} sin (πn x)

= \int_{0}^{1} x sin (πn x) - x^{2} sin (πn x) 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} x sin (πn x) d x - \int_{0}^{1} x^{2} sin (πn x) d x

\int_{0}^{1} x sin (πn x) d x = - \frac{( - 1 ) ^{n}}{πn}

\int_{0}^{1} x^{2} sin (πn x) d x = \frac{2 ( - 1 ) ^{n} - 2}{π ^{3} n ^{3}} - \frac{( - 1 ) ^{n}}{πn}

= - \frac{( - 1 ) ^{n}}{πn} - (\frac{2 ( - 1 ) ^{n} - 2}{π ^{3} n ^{3}} - \frac{( - 1 ) ^{n}}{πn})

Vereinfache

= - \frac{2 ( - 1 ) ^{n} - 2}{π ^{3} n ^{3}}

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

\int_{- 1}^{0} 9 x^{2} (2 x^{3} + 5)^{2} d x

\int_{1}^{4} x^{- 5} d x

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\int_{3}^{6} ∣ x - 4 ∣ d x