de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

= \int - \frac{3}{4} - \frac{1}{2 x} + \frac{2 x + 3}{8 ( x ^{2} + 1 )} + \frac{5}{16 ( x + 1 )} - \frac{1}{16 ( x - 1 )} 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{3}{4} d x - \int \frac{1}{2 x} d x + \int \frac{2 x + 3}{8 ( x ^{2} + 1 )} d x + \int \frac{5}{16 ( x + 1 )} d x - \int \frac{1}{16 ( x - 1 )} d x

\int \frac{3}{4} d x = \frac{3}{4} x

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

\int \frac{2 x + 3}{8 ( x ^{2} + 1 )} d x = \frac{1}{8} (ln x^{2} + 1 + 3 arctan (x))

\int \frac{5}{16 ( x + 1 )} d x = \frac{5}{16} ln ∣ x + 1 ∣

\int \frac{1}{16 ( x - 1 )} d x = \frac{1}{16} ln ∣ x - 1 ∣

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

t a y l or \frac{3}{x}, - 1

derivative of 9sin(5x)

\frac{d}{d x} (9 sin (5 x))

derivative g(t)=2log_{5}(t)

d er i v a t i v e g (t) = 2 lo g_{5} (t)

integral from 0 to 1 of 2/(sqrt(4-x^2))

\int_{0}^{1} \frac{2}{4 - x ^{2}} d x

(\partial)/(\partial x)(x^{0.25}y^{0.25})

\frac{\partial}{\partial x} (x^{0.25} y^{0.25})