integral from 1/2 to 1/3 of (cos(pix))

Lösung

\int_{\frac{1}{2}}^{\frac{1}{3}} (cos (π x)) d x

- \frac{2 - 3}{2 π}

Dezimale

- 0.04264 \dots

Schritte zur Lösung

\int_{\frac{1}{2}}^{\frac{1}{3}} cos (π x) d x

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{\frac{1}{3}}^{\frac{1}{2}} cos (π x) d x

Wende U-Substitution an

= - \int_{\frac{π}{3}}^{\frac{π}{2}} cos (u) \frac{1}{π} d u

Entferne die Konstante:

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

= - \frac{1}{π} \cdot \int_{\frac{π}{3}}^{\frac{π}{2}} cos (u) d u

Nutze das gemeinsame Integral :

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

= - \frac{1}{π} [sin (u)]_{\frac{π}{3}}^{\frac{π}{2}}

Berechne die Grenzen:

1 - \frac{3}{2}

= - \frac{1}{π} (1 - \frac{3}{2})

Vereinfache

= - \frac{2 - 3}{2 π}

\int_{0}^{1} 2 π x 1 + 4 x^{2} d x

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

\int_{0}^{8} \frac{3 x}{1 + x} d x

\int_{0}^{2 π} 3 sin (x) d x

\int_{1}^{5} ∣ 4 x - 8 ∣ d x