integral of (5x+3)/(x^2+3x+4)

Lösung

\int \frac{5 x + 3}{x ^{2} + 3 x + 4} d x

4 (5 (\frac{1}{8} ln 4 x^{2} + 12 x + 16 - \frac{3}{4 7} arctan (\frac{2 x + 3}{7})) + \frac{3}{2 7} arctan (\frac{2 x + 3}{7})) + C

Schritte zur Lösung

\int \frac{5 x + 3}{x ^{2} + 3 x + 4} d x

Vervollständige das Quadrat

x^{2} + 3 x + 4 : (x + \frac{3}{2})^{2} + \frac{7}{4}

= \int \frac{5 x + 3}{( x + \frac{3}{2} ) ^{2} + \frac{7}{4}} d x

Wende U-Substitution an

= \int \frac{4 ( 5 ( u - \frac{3}{2} ) + 3 )}{4 u ^{2} + 7} d u

Entferne die Konstante:

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

= 4 \cdot \int \frac{5 ( u - \frac{3}{2} ) + 3}{4 u ^{2} + 7} d u

Multipliziere aus

\frac{5 ( u - \frac{3}{2} ) + 3}{4 u ^{2} + 7} : \frac{5 ( u - \frac{3}{2} )}{4 u ^{2} + 7} + \frac{3}{4 u ^{2} + 7}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 (\int \frac{5 ( u - \frac{3}{2} )}{4 u ^{2} + 7} d u + \int \frac{3}{4 u ^{2} + 7} d u)

\int \frac{5 ( u - \frac{3}{2} )}{4 u ^{2} + 7} d u = 5 (\frac{1}{8} ln 4 u^{2} + 7 - \frac{3}{4 7} arctan (\frac{2}{7} u))

\int \frac{3}{4 u ^{2} + 7} d u = \frac{3}{2 7} arctan (\frac{2}{7} u)

= 4 (5 (\frac{1}{8} ln 4 u^{2} + 7 - \frac{3}{4 7} arctan (\frac{2}{7} u)) + \frac{3}{2 7} arctan (\frac{2}{7} u))

Setze in

u = x + \frac{3}{2}

ein

= 4 (5 (\frac{1}{8} ln 4 (x + \frac{3}{2})^{2} + 7 - \frac{3}{4 7} arctan (\frac{2}{7} (x + \frac{3}{2}))) + \frac{3}{2 7} arctan (\frac{2}{7} (x + \frac{3}{2})))

Vereinfache

4 (5 (\frac{1}{8} ln 4 (x + \frac{3}{2})^{2} + 7 - \frac{3}{4 7} arctan (\frac{2}{7} (x + \frac{3}{2}))) + \frac{3}{2 7} arctan (\frac{2}{7} (x + \frac{3}{2}))) : 4 (5 (\frac{1}{8} ln 4 x^{2} + 12 x + 16 - \frac{3}{4 7} arctan (\frac{2 x + 3}{7})) + \frac{3}{2 7} arctan (\frac{2 x + 3}{7}))

= 4 (5 (\frac{1}{8} ln 4 x^{2} + 12 x + 16 - \frac{3}{4 7} arctan (\frac{2 x + 3}{7})) + \frac{3}{2 7} arctan (\frac{2 x + 3}{7}))

Füge eine Konstante zur Lösung hinzu

= 4 (5 (\frac{1}{8} ln 4 x^{2} + 12 x + 16 - \frac{3}{4 7} arctan (\frac{2 x + 3}{7})) + \frac{3}{2 7} arctan (\frac{2 x + 3}{7})) + C

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