integral from 0 to 1 of x*sin(npix)

Lösung

\int_{0}^{1} x \cdot sin (nπ x) d x

- \frac{( - 1 ) ^{n}}{πn}

Schritte zur Lösung

\int_{0}^{1} x sin (nπ x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

Berechne die Grenzen:

- (- 1)^{n} \frac{1}{πn}

= - (- 1)^{n} \frac{1}{πn}

Vereinfache

= - \frac{( - 1 ) ^{n}}{πn}

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

\int_{0}^{2} x^{3} - x^{2} - 2 x d x

\int_{1}^{5} - x^{2} + 6 x - 5 d x

\int_{0}^{l n (5)} 2 e^{3 x} d x

\int_{- 1}^{1} y^{3} - y d y