integral from x to 2 of 2y^2sin(xy)

Lösung

\int_{x}^{2} 2 y^{2} sin (x y) d y

2 (\frac{- 4 cos ( 2 x ) - 2 sin ( x ^{2} )}{x} + \frac{4 sin ( 2 x )}{x ^{2}} + \frac{2 cos ( 2 x ) - 2 cos ( x ^{2} )}{x ^{3}} + x cos (x^{2}))

Schritte zur Lösung

\int_{x}^{2} 2 y^{2} sin (x y) d y

Entferne die Konstante:

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

= 2 \cdot \int_{x}^{2} sin (x y) y^{2} d y

Wende die partielle Integration an

= 2 [- \frac{1}{x} y^{2} cos (x y) - \int - \frac{2}{x} y cos (x y) d y]_{x}^{2}

\int - \frac{2}{x} y cos (x y) d y = - \frac{2}{x ^{3}} (x y sin (x y) + cos (x y))

= 2 [- \frac{1}{x} y^{2} cos (x y) - (- \frac{2}{x ^{3}} (x y sin (x y) + cos (x y)))]_{x}^{2}

Vereinfache

= 2 [- \frac{1}{x} y^{2} cos (x y) + \frac{2}{x ^{3}} (x y sin (x y) + cos (x y))]_{x}^{2}

Berechne die Grenzen:

- \frac{4}{x} cos (2 x) + \frac{4}{x ^{2}} sin (2 x) + \frac{2}{x ^{3}} cos (2 x) + x cos (x^{2}) - \frac{2}{x} sin (x^{2}) - \frac{2}{x ^{3}} cos (x^{2})

= 2 (- \frac{4}{x} cos (2 x) + \frac{4}{x ^{2}} sin (2 x) + \frac{2}{x ^{3}} cos (2 x) + x cos (x^{2}) - \frac{2}{x} sin (x^{2}) - \frac{2}{x ^{3}} cos (x^{2}))

Vereinfache

= 2 (\frac{- 4 cos ( 2 x ) - 2 sin ( x ^{2} )}{x} + \frac{4 sin ( 2 x )}{x ^{2}} + \frac{2 cos ( 2 x ) - 2 cos ( x ^{2} )}{x ^{3}} + x cos (x^{2}))

\int_{2}^{4} (x^{2} - 2 x - 3) d x

\int_{0}^{1} x 3 - 2 x^{2} d x

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

\int_{0}^{\frac{π}{2}} 3 sin (y) cos (y) d y

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