integral of (5x+6)/((x-2)(3x+1)^2)

Lösung

\int \frac{5 x + 6}{( x - 2 ) ( 3 x + 1 ) ^{2}} d x

\frac{16}{49} ln ∣ x - 2 ∣ - \frac{16}{49} ln ∣ 3 x + 1 ∣ + \frac{13}{21 ( 3 x + 1 )} + C

Schritte zur Lösung

\int \frac{5 x + 6}{( x - 2 ) ( 3 x + 1 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{5 x + 6}{( x - 2 ) ( 3 x + 1 ) ^{2}} : \frac{16}{49 ( x - 2 )} - \frac{48}{49 ( 3 x + 1 )} - \frac{13}{7 ( 3 x + 1 ) ^{2}}

= \int \frac{16}{49 ( x - 2 )} - \frac{48}{49 ( 3 x + 1 )} - \frac{13}{7 ( 3 x + 1 ) ^{2}} 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{16}{49 ( x - 2 )} d x - \int \frac{48}{49 ( 3 x + 1 )} d x - \int \frac{13}{7 ( 3 x + 1 ) ^{2}} d x

\int \frac{16}{49 ( x - 2 )} d x = \frac{16}{49} ln ∣ x - 2 ∣

\int \frac{48}{49 ( 3 x + 1 )} d x = \frac{16}{49} ln ∣ 3 x + 1 ∣

\int \frac{13}{7 ( 3 x + 1 ) ^{2}} d x = - \frac{13}{21 ( 3 x + 1 )}

= \frac{16}{49} ln ∣ x - 2 ∣ - \frac{16}{49} ln ∣ 3 x + 1 ∣ - (- \frac{13}{21 ( 3 x + 1 )})

Vereinfache

= \frac{16}{49} ln ∣ x - 2 ∣ - \frac{16}{49} ln ∣ 3 x + 1 ∣ + \frac{13}{21 ( 3 x + 1 )}

Füge eine Konstante zur Lösung hinzu

= \frac{16}{49} ln ∣ x - 2 ∣ - \frac{16}{49} ln ∣ 3 x + 1 ∣ + \frac{13}{21 ( 3 x + 1 )} + C

d er i v a t i v e f (y) = e^{t a n (x)}

\frac{d y}{d x} = \frac{y + 4}{2 - y}

\frac{d q}{d t} = k (q - 70)

\frac{d}{d x} (6^{- x})

d er i v a t i v e x^{3}