integral from 0 to l of xcos((npix)/l)

Lösung

\int_{0}^{l} x cos (\frac{nπ x}{l}) d x

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

Schritte zur Lösung

\int_{0}^{l} x cos (\frac{nπ x}{l}) d x

Wende die partielle Integration an

= [l \frac{1}{πn} x sin (\frac{πn x}{l}) - \int l \frac{1}{πn} sin (\frac{πn x}{l}) d x]_{0}^{l}

\int l \frac{1}{πn} sin (\frac{πn x}{l}) d x = - l \frac{l}{π ^{2} n ^{2}} cos (\frac{πn x}{l})

= [l \frac{1}{πn} x sin (\frac{πn x}{l}) - (- l \frac{l}{π ^{2} n ^{2}} cos (\frac{πn x}{l}))]_{0}^{l}

Vereinfache

= [l \frac{1}{πn} x sin (\frac{πn x}{l}) + l \frac{l}{π ^{2} n ^{2}} cos (\frac{πn x}{l})]_{0}^{l}

Berechne die Grenzen:

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

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

Vereinfache

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

\int_{0}^{5} 8 x^{3} d x

\int_{0}^{1} \frac{1}{2 x + 4} d x

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

\int_{0}^{2 π} - 6 sin^{2} (t) d t

\int_{0}^{π} x cos (\frac{n x}{2}) d x