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Beliebt Rechnen >

integral of (t^2-2)/((t^2-t-6)^2)

Lösung

\int \frac{t ^{2} - 2}{( t ^{2} - t - 6 ) ^{2}} d t

Lösung

- \frac{16}{125} ln ∣ t + 2 ∣ - \frac{2}{25 ( t + 2 )} + \frac{16}{125} ln ∣ t - 3 ∣ - \frac{7}{25 ( t - 3 )} + C

Schritte zur Lösung

\int \frac{t ^{2} - 2}{( t ^{2} - t - 6 ) ^{2}} d t

Ermittle den Partialbruch von

\frac{t ^{2} - 2}{( t ^{2} - t - 6 ) ^{2}} : - \frac{16}{125 ( t + 2 )} + \frac{2}{25 ( t + 2 ) ^{2}} + \frac{16}{125 ( t - 3 )} + \frac{7}{25 ( t - 3 ) ^{2}}

= \int - \frac{16}{125 ( t + 2 )} + \frac{2}{25 ( t + 2 ) ^{2}} + \frac{16}{125 ( t - 3 )} + \frac{7}{25 ( t - 3 ) ^{2}} d t

Wende die Summenregel an:

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

= - \int \frac{16}{125 ( t + 2 )} d t + \int \frac{2}{25 ( t + 2 ) ^{2}} d t + \int \frac{16}{125 ( t - 3 )} d t + \int \frac{7}{25 ( t - 3 ) ^{2}} d t

\int \frac{16}{125 ( t + 2 )} d t = \frac{16}{125} ln ∣ t + 2 ∣

\int \frac{2}{25 ( t + 2 ) ^{2}} d t = - \frac{2}{25 ( t + 2 )}

\int \frac{16}{125 ( t - 3 )} d t = \frac{16}{125} ln ∣ t - 3 ∣

\int \frac{7}{25 ( t - 3 ) ^{2}} d t = - \frac{7}{25 ( t - 3 )}

= - \frac{16}{125} ln ∣ t + 2 ∣ - \frac{2}{25 ( t + 2 )} + \frac{16}{125} ln ∣ t - 3 ∣ - \frac{7}{25 ( t - 3 )}

Füge eine Konstante zur Lösung hinzu

= - \frac{16}{125} ln ∣ t + 2 ∣ - \frac{2}{25 ( t + 2 )} + \frac{16}{125} ln ∣ t - 3 ∣ - \frac{7}{25 ( t - 3 )} + C

Graph

Beliebte Beispiele

integral of ((x^3))/(sqrt(x^2+49))

\int \frac{( x ^{3} )}{x ^{2} + 49} d x

integral of ((x+1)^2)/(sqrt(x))

\int \frac{( x + 1 ) ^{2}}{x} d x

integral of (4x)/(sqrt(x-5))

\int \frac{4 x}{x - 5} d x

integral of-(cos(x))/(e^x)

\int - \frac{cos ( x )}{e ^{x}} d x

maclaurin 1/(1+x)

ma c l a u r in \frac{1}{1 + x}