integral of (x^2+1)/((x-7)(x-6)^2)

Lösung

\int \frac{x ^{2} + 1}{( x - 7 ) ( x - 6 ) ^{2}} d x

50 ln ∣ x - 7 ∣ - 49 ln ∣ x - 6 ∣ + \frac{37}{x - 6} + C

Schritte zur Lösung

\int \frac{x ^{2} + 1}{( x - 7 ) ( x - 6 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{x ^{2} + 1}{( x - 7 ) ( x - 6 ) ^{2}} : \frac{50}{x - 7} - \frac{49}{x - 6} - \frac{37}{( x - 6 ) ^{2}}

= \int \frac{50}{x - 7} - \frac{49}{x - 6} - \frac{37}{( x - 6 ) ^{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{50}{x - 7} d x - \int \frac{49}{x - 6} d x - \int \frac{37}{( x - 6 ) ^{2}} d x

\int \frac{50}{x - 7} d x = 50 ln ∣ x - 7 ∣

\int \frac{49}{x - 6} d x = 49 ln ∣ x - 6 ∣

\int \frac{37}{( x - 6 ) ^{2}} d x = - \frac{37}{x - 6}

= 50 ln ∣ x - 7 ∣ - 49 ln ∣ x - 6 ∣ - (- \frac{37}{x - 6})

Vereinfache

= 50 ln ∣ x - 7 ∣ - 49 ln ∣ x - 6 ∣ + \frac{37}{x - 6}

Füge eine Konstante zur Lösung hinzu

= 50 ln ∣ x - 7 ∣ - 49 ln ∣ x - 6 ∣ + \frac{37}{x - 6} + C

y^{^{'}} = - 0.5 y + x

\int \frac{y}{2} d y

d er i v a t i v e f (x) = 5 x 2

\int 17 (tan (x)) (ln (cos (x))) d x

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