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Beliebt Rechnen >

integral of 1/((t+2)^2(t+1))

Lösung

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

Lösung

- ln ∣ t + 2 ∣ + \frac{1}{t + 2} + ln ∣ t + 1 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{( t + 2 ) ^{2} ( t + 1 )} : - \frac{1}{t + 2} - \frac{1}{( t + 2 ) ^{2}} + \frac{1}{t + 1}

= \int - \frac{1}{t + 2} - \frac{1}{( t + 2 ) ^{2}} + \frac{1}{t + 1} 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{1}{t + 2} d t - \int \frac{1}{( t + 2 ) ^{2}} d t + \int \frac{1}{t + 1} d t

\int \frac{1}{t + 2} d t = ln ∣ t + 2 ∣

\int \frac{1}{( t + 2 ) ^{2}} d t = - \frac{1}{t + 2}

\int \frac{1}{t + 1} d t = ln ∣ t + 1 ∣

= - ln ∣ t + 2 ∣ - (- \frac{1}{t + 2}) + ln ∣ t + 1 ∣

Vereinfache

= - ln ∣ t + 2 ∣ + \frac{1}{t + 2} + ln ∣ t + 1 ∣

Füge eine Konstante zur Lösung hinzu

= - ln ∣ t + 2 ∣ + \frac{1}{t + 2} + ln ∣ t + 1 ∣ + C

Graph

Beliebte Beispiele

integral of 6/(x+\sqrt[3]{x)}

\int \frac{6}{x + 3 x} d x

integral of 5sinh(x/2-ln(3))

\int 5 sinh (\frac{x}{2} - ln (3)) d x

integral of (ln(x)+1)/(x*ln(x))

\int \frac{ln ( x ) + 1}{x \cdot ln ( x )} d x

integral of sec^3(y)

\int sec^{3} (y) d y

integral of ((2x+7))/(x^2+2x+5)

\int \frac{( 2 x + 7 )}{x ^{2} + 2 x + 5} d x