integral of 1/(t^{1/2)-t^{1/4}}

Lösung

\int \frac{1}{t ^{\frac{1}{2}} - t ^{\frac{1}{4}}} d t

4 \frac{( t ^{\frac{1}{4}} - 1 ) ^{2}}{2} + 2 (t^{\frac{1}{4}} - 1) + ln t^{\frac{1}{4}} - 1 + C

Schritte zur Lösung

\int \frac{1}{t ^{\frac{1}{2}} - t ^{\frac{1}{4}}} d t

Wende U-Substitution an

= \int \frac{4 u ^{2}}{u - 1} d u

Entferne die Konstante:

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

= 4 \cdot \int \frac{u ^{2}}{u - 1} d u

Wende U-Substitution an

= 4 \cdot \int \frac{( v + 1 ) ^{2}}{v} d v

Multipliziere aus

\frac{( v + 1 ) ^{2}}{v} : v + 2 + \frac{1}{v}

= 4 \cdot \int v + 2 + \frac{1}{v} d v

Wende die Summenregel an:

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

= 4 (\int v d v + \int 2 d v + \int \frac{1}{v} d v)

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

\int 2 d v = 2 v

\int \frac{1}{v} d v = ln ∣ v ∣

= 4 (\frac{v ^{2}}{2} + 2 v + ln ∣ v ∣)

Ersetze zurück

= 4 \frac{( t ^{\frac{1}{4}} - 1 ) ^{2}}{2} + 2 (t^{\frac{1}{4}} - 1) + ln t^{\frac{1}{4}} - 1

Füge eine Konstante zur Lösung hinzu

= 4 \frac{( t ^{\frac{1}{4}} - 1 ) ^{2}}{2} + 2 (t^{\frac{1}{4}} - 1) + ln t^{\frac{1}{4}} - 1 + C

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