integral from 0 to y^2 of ycos(x/y)

Lösung

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

y^{2} sin (y)

Schritte zur Lösung

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

Entferne die Konstante:

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

= y \cdot \int_{0}^{y^{2}} cos (\frac{x}{y}) d x

Wende U-Substitution an

= y \cdot \int_{0}^{y} cos (u) y d u

Entferne die Konstante:

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

= yy \cdot \int_{0}^{y} cos (u) d u

Nutze das gemeinsame Integral :

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

= yy [sin (u)]_{0}^{y}

Vereinfache

yy [sin (u)]_{0}^{y} : y^{2} [sin (u)]_{0}^{y}

= y^{2} [sin (u)]_{0}^{y}

Berechne die Grenzen:

sin (y)

= y^{2} sin (y)

\int_{1}^{5} (\frac{x ^{2} + 3}{x}) d x

\int_{1}^{4} x ln (x^{9}) d x

\int_{0}^{x^{2}} 1 + x^{2} d x

\int_{4}^{8} \frac{3}{x ^{3}} d x

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