integral of (x^2)/(sqrt(20-4x+x^2))

Lösung

\int \frac{x ^{2}}{20 - 4 x + x ^{2}} d x

- 4 ln (\frac{1}{4} x + x^{2} - 4 x + 20 - 2) + \frac{1}{2} (x - 2) x^{2} - 4 x + 20 + 4 x^{2} - 4 x + 20 + C

Schritte zur Lösung

\int \frac{x ^{2}}{20 - 4 x + x ^{2}} d x

Vervollständige das Quadrat

20 - 4 x + x^{2} : (x - 2)^{2} + 16

= \int \frac{x ^{2}}{( x - 2 ) ^{2} + 16} d x

Wende U-Substitution an

= \int \frac{( u + 2 ) ^{2}}{u ^{2} + 16} d u

Multipliziere aus

\frac{( u + 2 ) ^{2}}{u ^{2} + 16} : \frac{u ^{2}}{u ^{2} + 16} + \frac{4 u}{u ^{2} + 16} + \frac{4}{u ^{2} + 16}

= \int \frac{u ^{2}}{u ^{2} + 16} + \frac{4 u}{u ^{2} + 16} + \frac{4}{u ^{2} + 16} d u

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{u ^{2}}{u ^{2} + 16} d u + \int \frac{4 u}{u ^{2} + 16} d u + \int \frac{4}{u ^{2} + 16} d u

\int \frac{u ^{2}}{u ^{2} + 16} d u = 16 (- \frac{1}{2} ln (\frac{u + 16 + u ^{2}}{4}) + \frac{1}{32} u 16 + u^{2})

\int \frac{4 u}{u ^{2} + 16} d u = 4 u^{2} + 16

\int \frac{4}{u ^{2} + 16} d u = 4 ln (\frac{1}{4} u + 16 + u^{2})

= 16 (- \frac{1}{2} ln (\frac{u + 16 + u ^{2}}{4}) + \frac{1}{32} u 16 + u^{2}) + 4 u^{2} + 16 + 4 ln (\frac{1}{4} u + 16 + u^{2})

Setze in

u = x - 2

ein

= 16 - \frac{1}{2} ln \frac{x - 2 + 16 + ( x - 2 ) ^{2}}{4} + \frac{1}{32} (x - 2) 16 + (x - 2)^{2} + 4 (x - 2)^{2} + 16 + 4 ln (\frac{1}{4} x - 2 + 16 + (x - 2)^{2})

Vereinfache

16 - \frac{1}{2} ln \frac{x - 2 + 16 + ( x - 2 ) ^{2}}{4} + \frac{1}{32} (x - 2) 16 + (x - 2)^{2} + 4 (x - 2)^{2} + 16 + 4 ln (\frac{1}{4} x - 2 + 16 + (x - 2)^{2}) : - 4 ln (\frac{1}{4} x + x^{2} - 4 x + 20 - 2) + \frac{1}{2} (x - 2) x^{2} - 4 x + 20 + 4 x^{2} - 4 x + 20

= - 4 ln (\frac{1}{4} x + x^{2} - 4 x + 20 - 2) + \frac{1}{2} (x - 2) x^{2} - 4 x + 20 + 4 x^{2} - 4 x + 20

Füge eine Konstante zur Lösung hinzu

= - 4 ln (\frac{1}{4} x + x^{2} - 4 x + 20 - 2) + \frac{1}{2} (x - 2) x^{2} - 4 x + 20 + 4 x^{2} - 4 x + 20 + C

\frac{d}{d x} (0.5 sin^{2} (x) - x + 1)

in v erse l a pl a ce \frac{4}{( s + 1 ) ^{2}}

t \cdot y^{^{'}} + t y = 1 - y

l a pl a ce t r an s f or m t^{2} \cdot e^{a \cdot t}

t^{2} y^{^{''}} + 19 t y^{^{'}} + 81 y = 0