integral of xsqrt(x^4+81)

Lösung

\int x x^{4} + 81 d x

\frac{1}{4} (x^{2} x^{4} + 81 + 81 ln (\frac{x ^{2} + x ^{4} + 81}{9})) + C

Schritte zur Lösung

\int x x^{4} + 81 d x

Vereinfache

x^{4} : (x^{2})^{2}

= \int x (x^{2})^{2} + 81 d x

Wende U-Substitution an

= \int \frac{u ^{2} + 81}{2} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int u^{2} + 81 d u

Trigonometrische Substitution anwenden

= \frac{1}{2} \cdot \int 81 sec^{3} (v) d v

Entferne die Konstante:

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

= \frac{1}{2} \cdot 81 \cdot \int sec^{3} (v) d v

Wende integrale Reduktion an

= \frac{1}{2} \cdot 81 (\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{1}{2} \cdot 81 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Ersetze zurück

= \frac{1}{2} \cdot 81 (\frac{sec ^{2} ( arctan ( \frac{1}{9} x ^{2} ) ) sin ( arctan ( \frac{1}{9} x ^{2} ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{9} x^{2})) + sec (arctan (\frac{1}{9} x^{2})))

Vereinfache

\frac{1}{2} \cdot 81 (\frac{sec ^{2} ( arctan ( \frac{1}{9} x ^{2} ) ) sin ( arctan ( \frac{1}{9} x ^{2} ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{9} x^{2})) + sec (arctan (\frac{1}{9} x^{2}))) : \frac{1}{4} (x^{2} x^{4} + 81 + 81 ln (\frac{x ^{2} + x ^{4} + 81}{9}))

= \frac{1}{4} (x^{2} x^{4} + 81 + 81 ln (\frac{x ^{2} + x ^{4} + 81}{9}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} (x^{2} x^{4} + 81 + 81 ln (\frac{x ^{2} + x ^{4} + 81}{9})) + C

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