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Beliebt Rechnen >

integral from 0 to 1 of t^2cos(npit)

Lösung

\int_{0}^{1} t^{2} cos (nπ t) d t

Lösung

\frac{2 ( - 1 ) ^{n}}{π ^{2} n ^{2}}

Schritte zur Lösung

\int_{0}^{1} t^{2} cos (nπ t) d t

Wende U-Substitution an

= \int_{0}^{πn} \frac{u ^{2} cos ( u )}{π ^{3} n ^{3}} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{π ^{3} n ^{3}} \cdot \int_{0}^{πn} u^{2} cos (u) d u

Wende die partielle Integration an

= \frac{1}{π ^{3} n ^{3}} [u^{2} sin (u) - \int 2 u sin (u) d u]_{0}^{πn}

\int 2 u sin (u) d u = 2 (- u cos (u) + sin (u))

= \frac{1}{π ^{3} n ^{3}} [u^{2} sin (u) - 2 (- u cos (u) + sin (u))]_{0}^{πn}

Berechne die Grenzen:

2 π (- 1)^{n} n

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

Vereinfache

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

Beliebte Beispiele

(dy)/(dx)=sin^2(x)

\frac{d y}{d x} = sin^{2} (x)

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

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\frac{d}{d x} (\frac{3}{( x + 2 ) ^{2}})

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\frac{d}{d x} (2 cot (\frac{x}{2}))