integral from 1 to 2 of sin((npix)/2)

Lösung

\int_{1}^{2} sin (\frac{nπ x}{2}) d x

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

Schritte zur Lösung

\int_{1}^{2} sin (\frac{nπ x}{2}) d x

\frac{nπ x}{2} = \frac{nπ}{2} x

= \int_{1}^{2} sin (\frac{nπ}{2} x) d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

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

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

Berechne die Grenzen:

- (- 1)^{n} + cos (\frac{πn}{2})

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

Vereinfache

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

\frac{d}{d x} (4 (x + 2)^{3})

x \to - 3 lim (- (\frac{1}{x ^{2} - 9}))

\int \frac{3 x ^{3} + 6 x ^{2} + 3}{3 x + 3} d x

x \to \infty lim (0.6 6^{x})

\int 3 x - 6 x d x