integral of (33x^2)/(sqrt(x^2-4))

Lösung

\int \frac{33 x ^{2}}{x ^{2} - 4} d x

132 (\frac{1}{8} x x^{2} - 4 + \frac{1}{2} ln (\frac{x ^{2} - 4 + x}{2})) + C

Schritte zur Lösung

\int \frac{33 x ^{2}}{x ^{2} - 4} d x

Entferne die Konstante:

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

= 33 \cdot \int \frac{x ^{2}}{x ^{2} - 4} d x

Trigonometrische Substitution anwenden

= 33 \cdot \int 4 sec^{3} (u) d u

Entferne die Konstante:

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

= 33 \cdot 4 \cdot \int sec^{3} (u) d u

Wende integrale Reduktion an

= 33 \cdot 4 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

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

= 33 \cdot 4 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Setze in

u = arcsec (\frac{1}{2} x)

ein

= 33 \cdot 4 (\frac{sec ^{2} ( arcsec ( \frac{1}{2} x ) ) sin ( arcsec ( \frac{1}{2} x ) )}{2} + \frac{1}{2} ln tan (arcsec (\frac{1}{2} x)) + sec (arcsec (\frac{1}{2} x)))

Vereinfache

33 \cdot 4 (\frac{sec ^{2} ( arcsec ( \frac{1}{2} x ) ) sin ( arcsec ( \frac{1}{2} x ) )}{2} + \frac{1}{2} ln tan (arcsec (\frac{1}{2} x)) + sec (arcsec (\frac{1}{2} x))) : 132 (\frac{1}{8} x x^{2} - 4 + \frac{1}{2} ln (\frac{x ^{2} - 4 + x}{2}))

= 132 (\frac{1}{8} x x^{2} - 4 + \frac{1}{2} ln (\frac{x ^{2} - 4 + x}{2}))

Füge eine Konstante zur Lösung hinzu

= 132 (\frac{1}{8} x x^{2} - 4 + \frac{1}{2} ln (\frac{x ^{2} - 4 + x}{2})) + C

\int_{0}^{π} x cos^{2} (x) d x

x \to 0 lim ((1 - 7 x)^{\frac{1}{x}})

\int \frac{( x ^{3} - 4 x )}{( x ^{2} + 1 ) ^{2}} d x

\int 7 x^{4} - 5 x^{3} + 4 x^{2} + C d x

x \to \infty + lim (\frac{1}{x})