integral ((z+4))/((z^2+2z+5))

Lösung

integral

\frac{( z + 4 )}{( z ^{2} + 2 z + 5 )}

\frac{1}{2} ln z^{2} + 2 z + 5 + \frac{3}{2} arctan (\frac{1}{2} (z + 1)) + C

Schritte zur Lösung

\int \frac{z + 4}{z ^{2} + 2 z + 5} d z

Vervollständige das Quadrat

z^{2} + 2 z + 5 : (z + 1)^{2} + 4

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

Wende U-Substitution an

= \int \frac{u + 3}{u ^{2} + 4} d u

Multipliziere aus

\frac{u + 3}{u ^{2} + 4} : \frac{u}{u ^{2} + 4} + \frac{3}{u ^{2} + 4}

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}{u ^{2} + 4} d u + \int \frac{3}{u ^{2} + 4} d u

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

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

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

Setze in

u = z + 1

ein

= \frac{1}{2} ln (z + 1)^{2} + 4 + \frac{3}{2} arctan (\frac{z + 1}{2})

Vereinfache

\frac{1}{2} ln (z + 1)^{2} + 4 + \frac{3}{2} arctan (\frac{z + 1}{2}) : \frac{1}{2} ln z^{2} + 2 z + 5 + \frac{3}{2} arctan (\frac{1}{2} (z + 1))

= \frac{1}{2} ln z^{2} + 2 z + 5 + \frac{3}{2} arctan (\frac{1}{2} (z + 1))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln z^{2} + 2 z + 5 + \frac{3}{2} arctan (\frac{1}{2} (z + 1)) + C

in t e g r a l (3 x - 4)^{3}

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