integral of (3x+2)/(sqrt(x^2+2x+3))

Lösung

\int \frac{3 x + 2}{x ^{2} + 2 x + 3} d x

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

Schritte zur Lösung

\int \frac{3 x + 2}{x ^{2} + 2 x + 3} d x

Multipliziere aus

\frac{3 x + 2}{x ^{2} + 2 x + 3} : \frac{3 x}{x ^{2} + 2 x + 3} + \frac{2}{x ^{2} + 2 x + 3}

= \int \frac{3 x}{x ^{2} + 2 x + 3} + \frac{2}{x ^{2} + 2 x + 3} d x

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{3 x}{x ^{2} + 2 x + 3} d x + \int \frac{2}{x ^{2} + 2 x + 3} d x

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{( x + 2 )}{( x ^{2} + 4 x + 5 )} d x

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\int \frac{25}{x ^{2} - 25} d x