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Beliebt Rechnen >

integral of (4(x^2+1))/(x(x^2+4))

Lösung

\int \frac{4 ( x ^{2} + 1 )}{x ( x ^{2} + 4 )} d x

Lösung

\frac{3}{2} ln x^{2} + 4 + ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{4 ( x ^{2} + 1 )}{x ( x ^{2} + 4 )} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int \frac{x ^{2} + 1}{x ( x ^{2} + 4 )} d x

Multipliziere aus

\frac{x ^{2} + 1}{x ( x ^{2} + 4 )} : \frac{x ^{2}}{x ( x ^{2} + 4 )} + \frac{1}{x ( x ^{2} + 4 )}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 (\int \frac{x ^{2}}{x ( x ^{2} + 4 )} d x + \int \frac{1}{x ( x ^{2} + 4 )} d x)

\int \frac{x ^{2}}{x ( x ^{2} + 4 )} d x = \frac{1}{2} ln x^{2} + 4

\int \frac{1}{x ( x ^{2} + 4 )} d x = \frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4

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

Vereinfache

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

= \frac{3}{2} ln x^{2} + 4 + ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{3}{2} ln x^{2} + 4 + ln ∣ x ∣ + C

Graph

Beliebte Beispiele

integral of 1/(cos(5x))

\int \frac{1}{cos ( 5 x )} d x

integral of 2/(4x)

\int \frac{2}{4 x} d x

integral of (2y+3)/((y-1)(y^3-1))

\int \frac{2 y + 3}{( y - 1 ) ( y ^{3} - 1 )} d y

integral of t(sec^2(t^2))(tan^4(t^2))

\int t (sec^{2} (t^{2})) (tan^{4} (t^{2})) d t

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

\int \frac{1}{x ^{2}} - 6 x d x