integral from pi to 2pi of sin((nx)/2)

Lösung

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

\frac{2 ( - ( - 1 ) ^{n} + cos ( \frac{πn}{2} ) )}{n}

Schritte zur Lösung

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

Wende U-Substitution an

= \int_{\frac{nπ}{2}}^{πn} \frac{2 sin ( u )}{n} d u

Entferne die Konstante:

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

= \frac{2}{n} \cdot \int_{\frac{nπ}{2}}^{πn} sin (u) d u

Nutze das gemeinsame Integral :

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

= \frac{2}{n} [- cos (u)]_{\frac{nπ}{2}}^{πn}

Berechne die Grenzen:

- (- 1)^{n} + cos (\frac{πn}{2})

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

Vereinfache

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

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

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

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

\int_{1}^{x} \frac{e ^{t}}{t} d t

\int_{0}^{1} (1 - x)^{5} d x