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Beliebt Rechnen >

integral of 1/(x^4+x^3)

Lösung

\int \frac{1}{x ^{4} + x ^{3}} d x

Lösung

ln ∣ x ∣ + \frac{1}{x} - \frac{1}{2 x ^{2}} - ln ∣ x + 1 ∣ + C

Schritte zur Lösung

\int \frac{1}{x ^{4} + x ^{3}} d x

Ermittle den Partialbruch von

\frac{1}{x ^{4} + x ^{3}} : \frac{1}{x} - \frac{1}{x ^{2}} + \frac{1}{x ^{3}} - \frac{1}{x + 1}

= \int \frac{1}{x} - \frac{1}{x ^{2}} + \frac{1}{x ^{3}} - \frac{1}{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{1}{x} d x - \int \frac{1}{x ^{2}} d x + \int \frac{1}{x ^{3}} d x - \int \frac{1}{x + 1} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{1}{x ^{2}} d x = - \frac{1}{x}

\int \frac{1}{x ^{3}} d x = - \frac{1}{2 x ^{2}}

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

= ln ∣ x ∣ - (- \frac{1}{x}) - \frac{1}{2 x ^{2}} - ln ∣ x + 1 ∣

Vereinfache

= ln ∣ x ∣ + \frac{1}{x} - \frac{1}{2 x ^{2}} - ln ∣ x + 1 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ + \frac{1}{x} - \frac{1}{2 x ^{2}} - ln ∣ x + 1 ∣ + C

Graph

Beliebte Beispiele

integral of (ln(n^4))/n

\int \frac{ln ( n ^{4} )}{n}

integral of (110)/((x+1)(x^2+9)^2)

\int \frac{110}{( x + 1 ) ( x ^{2} + 9 ) ^{2}} d x

integral of (-3)/(sqrt(2t+3))

\int \frac{- 3}{2 t + 3} d t

integral of 1/(x^n(1+x^n)^{1/n)}

\int \frac{1}{x ^{n} ( 1 + x ^{n} ) ^{\frac{1}{n}}} d x

integral of y/(sqrt(16-9y^2))

\int \frac{y}{16 - 9 y ^{2}} d y