integral from 0 to pi of t^2cos(nt)

Lösung

\int_{0}^{π} t^{2} cos (n t) d t

\frac{2 π ( - 1 ) ^{n}}{n ^{2}}

Schritte zur Lösung

\int_{0}^{π} t^{2} cos (n t) d t

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

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

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

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

Berechne die Grenzen:

2 π (- 1)^{n} n

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

Vereinfache

= \frac{2 π ( - 1 ) ^{n}}{n ^{2}}

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