de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (5x+2)/(sqrt(x^2+2x+10))

Lösung

\int \frac{5 x + 2}{x ^{2} + 2 x + 10} d x

Lösung

5 x^{2} + 2 x + 10 - 3 ln (\frac{1}{3} x + 1 + x^{2} + 2 x + 10) + C

Schritte zur Lösung

\int \frac{5 x + 2}{x ^{2} + 2 x + 10} d x

Vervollständige das Quadrat

x^{2} + 2 x + 10 : (x + 1)^{2} + 9

= \int \frac{5 x + 2}{( x + 1 ) ^{2} + 9} d x

Wende U-Substitution an

= \int \frac{5 u - 3}{u ^{2} + 9} d u

Multipliziere aus

\frac{5 u - 3}{u ^{2} + 9} : \frac{5 u}{u ^{2} + 9} - \frac{3}{u ^{2} + 9}

= \int \frac{5 u}{u ^{2} + 9} - \frac{3}{u ^{2} + 9} 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{5 u}{u ^{2} + 9} d u - \int \frac{3}{u ^{2} + 9} d u

\int \frac{5 u}{u ^{2} + 9} d u = 5 u^{2} + 9

\int \frac{3}{u ^{2} + 9} d u = 3 ln (\frac{1}{3} u + 9 + u^{2})

= 5 u^{2} + 9 - 3 ln (\frac{1}{3} u + 9 + u^{2})

Setze in

u = x + 1

ein

= 5 (x + 1)^{2} + 9 - 3 ln (\frac{1}{3} x + 1 + 9 + (x + 1)^{2})

Vereinfache

5 (x + 1)^{2} + 9 - 3 ln (\frac{1}{3} x + 1 + 9 + (x + 1)^{2}) : 5 x^{2} + 2 x + 10 - 3 ln (\frac{1}{3} x + 1 + x^{2} + 2 x + 10)

= 5 x^{2} + 2 x + 10 - 3 ln (\frac{1}{3} x + 1 + x^{2} + 2 x + 10)

Füge eine Konstante zur Lösung hinzu

= 5 x^{2} + 2 x + 10 - 3 ln (\frac{1}{3} x + 1 + x^{2} + 2 x + 10) + C

Graph

Beliebte Beispiele

integral of 4/(x^2sqrt(x^2-16))

\int \frac{4}{x ^{2} x ^{2} - 16} d x

integral of xsin^3(x^2)

\int x sin^{3} (x^{2}) d x

integral of (cos(x))/((1+sin(x))^3)

\int \frac{cos ( x )}{( 1 + sin ( x ) ) ^{3}} d x

integral of cos^2(t)sin^5(t)

\int cos^{2} (t) sin^{5} (t) d t

integral of 3x^2(x^3-1)^4

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