integral of (1+ln(x))sqrt(1+(xln(x))^2)

Lösung

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

\frac{1}{2} x ln (x) 1 + x^{2} ln^{2} (x) + \frac{1}{2} ln x ln (x) + 1 + x^{2} ln^{2} (x) + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int 1 + u^{2} d u

Trigonometrische Substitution anwenden

= \int sec^{3} (v) d v

Wende integrale Reduktion an

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

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= \frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣

Ersetze zurück

= \frac{sec ^{2} ( arctan ( x ln ( x ) ) ) sin ( arctan ( x ln ( x ) ) )}{2} + \frac{1}{2} ln ∣ tan (arctan (x ln (x))) + sec (arctan (x ln (x))) ∣

Vereinfache

\frac{sec ^{2} ( arctan ( x ln ( x ) ) ) sin ( arctan ( x ln ( x ) ) )}{2} + \frac{1}{2} ln ∣ tan (arctan (x ln (x))) + sec (arctan (x ln (x))) ∣ : \frac{1}{2} x ln (x) 1 + x^{2} ln^{2} (x) + \frac{1}{2} ln x ln (x) + 1 + x^{2} ln^{2} (x)

= \frac{1}{2} x ln (x) 1 + x^{2} ln^{2} (x) + \frac{1}{2} ln x ln (x) + 1 + x^{2} ln^{2} (x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x ln (x) 1 + x^{2} ln^{2} (x) + \frac{1}{2} ln x ln (x) + 1 + x^{2} ln^{2} (x) + C

\int x \cdot 5^{- x^{2}} d x

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