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Beliebt Rechnen >

integral of (y^4+4y^2-1)/(y^3+4y)

Lösung

\int \frac{y ^{4} + 4 y ^{2} - 1}{y ^{3} + 4 y} d y

Lösung

\frac{y ^{2}}{2} - \frac{1}{4} ln ∣ y ∣ + \frac{1}{8} ln y^{2} + 4 + C

Schritte zur Lösung

\int \frac{y ^{4} + 4 y ^{2} - 1}{y ^{3} + 4 y} d y

Lange Division

\frac{y ^{4} + 4 y ^{2} - 1}{y ^{3} + 4 y} : y - \frac{1}{y ^{3} + 4 y}

= \int y - \frac{1}{y ^{3} + 4 y} d y

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int y d y - \int \frac{1}{y ^{3} + 4 y} d y

\int y d y = \frac{y ^{2}}{2}

\int \frac{1}{y ^{3} + 4 y} d y = \frac{1}{4} ln ∣ y ∣ - \frac{1}{8} ln y^{2} + 4

= \frac{y ^{2}}{2} - (\frac{1}{4} ln ∣ y ∣ - \frac{1}{8} ln y^{2} + 4)

Vereinfache

= \frac{y ^{2}}{2} - \frac{1}{4} ln ∣ y ∣ + \frac{1}{8} ln y^{2} + 4

Füge eine Konstante zur Lösung hinzu

= \frac{y ^{2}}{2} - \frac{1}{4} ln ∣ y ∣ + \frac{1}{8} ln y^{2} + 4 + C

Graph

Beliebte Beispiele

integral of (sqrt(x)-x/2+2/(sqrt(x)))

\int (x - \frac{x}{2} + \frac{2}{x}) d x

integral of (cos(x)sin(x))/(1+cos^4(x))

\int \frac{cos ( x ) sin ( x )}{1 + cos ^{4} ( x )} d x

integral of 2/(cos(θ)-1)

\int \frac{2}{cos ( θ ) - 1} d θ

integral of (x+7)/(x^2-8x-19)

\int \frac{x + 7}{x ^{2} - 8 x - 19} d x

integral of cos(7t)cos(3t)

\int cos (7 t) cos (3 t) d t