integral from-pi to 0 of (t-pi)cos(nt)

Lösung

\int_{- π}^{0} (t - π) cos (n t) d t

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

Schritte zur Lösung

\int_{- π}^{0} (t - π) cos (n t) d t

Wende die partielle Integration an

= [\frac{1}{n} sin (n t) (t - π) - \int \frac{1}{n} sin (n t) d t]_{- π}^{0}

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

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

Vereinfache

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

Berechne die Grenzen:

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

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

Vereinfache

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

\int_{0}^{π} 1 - sin (x) d x

\int_{- 1}^{3} (x^{2} - 1)^{2} d x

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

\int_{- 2}^{1} ∣ x - 1 ∣ d x

\int_{2}^{3} 1 + x^{2} d x