integral of (x^2-3x+7)/(x^3+x)

Lösung

\int \frac{x ^{2} - 3 x + 7}{x ^{3} + x} d x

- 3 ln x^{2} + 1 - 3 arctan (x) + 7 ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{x ^{2} - 3 x + 7}{x ^{3} + x} d x

Multipliziere aus

\frac{x ^{2} - 3 x + 7}{x ^{3} + x} : \frac{x ^{2}}{x ^{3} + x} - \frac{3 x}{x ^{3} + x} + \frac{7}{x ^{3} + 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{x ^{2}}{x ^{3} + x} d x - \int \frac{3 x}{x ^{3} + x} d x + \int \frac{7}{x ^{3} + x} d x

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

\int \frac{3 x}{x ^{3} + x} d x = 3 arctan (x)

\int \frac{7}{x ^{3} + x} d x = 7 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

= \frac{1}{2} ln x^{2} + 1 - 3 arctan (x) + 7 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

Vereinfache

\frac{1}{2} ln x^{2} + 1 - 3 arctan (x) + 7 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) : - 3 ln x^{2} + 1 - 3 arctan (x) + 7 ln ∣ x ∣

= - 3 ln x^{2} + 1 - 3 arctan (x) + 7 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= - 3 ln x^{2} + 1 - 3 arctan (x) + 7 ln ∣ x ∣ + C

d er i v a t i v e f (x) = - 12

\int x^{2} 9 + x^{3} d x

\int x (4 x^{2} + 5)^{7} d x

d er i v a t i v e f (t) = e^{t}

t an g e n t \frac{12}{( 2 sin ( x ) + 6 cos ( x ) ) ^{2}}