integral of (2x^2)/(sqrt((x-1)^2+4))

Lösung

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

- 2 ln (\frac{1}{2} x + x^{2} - 2 x + 5 - 1) + x x^{2} - 2 x + 5 + 3 x^{2} - 2 x + 5 + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende U-Substitution an

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

Multipliziere aus

\frac{( u + 1 ) ^{2}}{u ^{2} + 4} : \frac{u ^{2}}{u ^{2} + 4} + \frac{2 u}{u ^{2} + 4} + \frac{1}{u ^{2} + 4}

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

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

\int \frac{u ^{2}}{u ^{2} + 4} d u = 4 (- \frac{1}{2} ln (\frac{u + 4 + u ^{2}}{2}) + \frac{1}{8} u 4 + u^{2})

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

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

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

Setze in

u = x - 1

ein

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

Vereinfache

2 4 - \frac{1}{2} ln \frac{x - 1 + 4 + ( x - 1 ) ^{2}}{2} + \frac{1}{8} (x - 1) 4 + (x - 1)^{2} + 2 (x - 1)^{2} + 4 + ln (\frac{1}{2} x - 1 + 4 + (x - 1)^{2}) : - 2 ln (\frac{1}{2} x + x^{2} - 2 x + 5 - 1) + x x^{2} - 2 x + 5 + 3 x^{2} - 2 x + 5

= - 2 ln (\frac{1}{2} x + x^{2} - 2 x + 5 - 1) + x x^{2} - 2 x + 5 + 3 x^{2} - 2 x + 5

Füge eine Konstante zur Lösung hinzu

= - 2 ln (\frac{1}{2} x + x^{2} - 2 x + 5 - 1) + x x^{2} - 2 x + 5 + 3 x^{2} - 2 x + 5 + C

t an g e n t y = (x + 2) (x - 1) (x - 3)

\int t sec^{2} (9 t) d t

\frac{d}{d x} (- 5 x^{2} + 10)

\frac{d}{d x} (\frac{1}{x ^{n}})

d er i v a t i v e y = \frac{4 x ^{2} + 4 x + 8}{x}