integral from 0 to 1/2 of cos(npix)

Lösung

\int_{0}^{\frac{1}{2}} cos (nπ x) d x

\frac{sin ( \frac{πn}{2} )}{πn}

Schritte zur Lösung

\int_{0}^{\frac{1}{2}} cos (nπ x) d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

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

= \frac{1}{πn} [sin (u)]_{0}^{\frac{πn}{2}}

Berechne die Grenzen:

sin (\frac{πn}{2})

= \frac{1}{πn} sin (\frac{πn}{2})

Vereinfache

= \frac{sin ( \frac{πn}{2} )}{πn}

\int_{0}^{π} - 6 x^{2} cos (3 x) d x

\int_{0}^{2 π} cos (θ) + sin (θ) d θ

\int_{- 2}^{1} x^{2} ∣ x ∣ d x

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\int_{4}^{9} 2 x^{\frac{3}{2}} d x