integral of (4x^2+3)/(0.4x^3+0.9x)

Lösung

\int \frac{4 x ^{2} + 3}{0.4 x ^{3} + 0.9 x} d x

3.33333 \dots ln ∣ x ∣ + 3.33333 \dots ln 0.4 x^{2} + 0.9 + C

Schritte zur Lösung

\int \frac{4 x ^{2} + 3}{0.4 x ^{3} + 0.9 x} d x

Faktorisiere

\frac{4 x ^{2} + 3}{0.4 x ^{3} + 0.9 x}

= \int \frac{4 x ^{2} + 3}{x ( 0.4 x ^{2} + 0.9 )} d x

Ermittle den Partialbruch von

\frac{4 x ^{2} + 3}{x ( 0.4 x ^{2} + 0.9 )} : \frac{10}{3 x} + \frac{2.66666 \dots x}{0.4 x ^{2} + 0.9}

= \int \frac{10}{3 x} + \frac{2.66666 \dots x}{0.4 x ^{2} + 0.9} 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{10}{3 x} d x + \int \frac{2.66666 \dots x}{0.4 x ^{2} + 0.9} d x

\int \frac{10}{3 x} d x = \frac{10}{3} ln ∣ x ∣

\int \frac{2.66666 \dots x}{0.4 x ^{2} + 0.9} d x = 3.33333 \dots ln 0.4 x^{2} + 0.9

= \frac{10}{3} ln ∣ x ∣ + 3.33333 \dots ln 0.4 x^{2} + 0.9

Vereinfache

= 3.33333 \dots ln ∣ x ∣ + 3.33333 \dots ln 0.4 x^{2} + 0.9

Füge eine Konstante zur Lösung hinzu

= 3.33333 \dots ln ∣ x ∣ + 3.33333 \dots ln 0.4 x^{2} + 0.9 + C

\frac{d}{d t} (\frac{t ^{3} - 2 t ^{2} + 2 t - 4}{t ^{2} + 2})

d er i v a t i v e sin (θ)

y^{^{'}} = y^{2} + 3

\frac{d}{d x} (- x e^{- 2 x y})

x \to 2 lim (3 x + 1)