integral of (t+5)/(t^2+t-2)

Lösung

\int \frac{t + 5}{t ^{2} + t - 2} d t

2 ln ∣ t - 1 ∣ - ln ∣ t + 2 ∣ + C

Schritte zur Lösung

\int \frac{t + 5}{t ^{2} + t - 2} d t

Ermittle den Partialbruch von

\frac{t + 5}{t ^{2} + t - 2} : \frac{2}{t - 1} - \frac{1}{t + 2}

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

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

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

= 2 ln ∣ t - 1 ∣ - ln ∣ t + 2 ∣

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ t - 1 ∣ - ln ∣ t + 2 ∣ + C

\int + \frac{2 cot ( x ) - 3 sin ^{2} ( x )}{sin ( x )} d x

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