integral of (2x^4+5x^2+1)/(x^3+x)

Lösung

\int \frac{2 x ^{4} + 5 x ^{2} + 1}{x ^{3} + x} d x

x^{2} + ln x^{2} + 1 + ln ∣ x ∣ + 1 + C

Schritte zur Lösung

\int \frac{2 x ^{4} + 5 x ^{2} + 1}{x ^{3} + x} d x

Multipliziere aus

\frac{2 x ^{4} + 5 x ^{2} + 1}{x ^{3} + x} : \frac{2 x ^{4}}{x ^{3} + x} + \frac{5 x ^{2}}{x ^{3} + x} + \frac{1}{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{2 x ^{4}}{x ^{3} + x} d x + \int \frac{5 x ^{2}}{x ^{3} + x} d x + \int \frac{1}{x ^{3} + x} d x

\int \frac{2 x ^{4}}{x ^{3} + x} d x = x^{2} + 1 - ln x^{2} + 1

\int \frac{5 x ^{2}}{x ^{3} + x} d x = \frac{5}{2} ln x^{2} + 1

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

= x^{2} + 1 - ln x^{2} + 1 + \frac{5}{2} ln x^{2} + 1 + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1

Vereinfache

x^{2} + 1 - ln x^{2} + 1 + \frac{5}{2} ln x^{2} + 1 + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1 : x^{2} + ln x^{2} + 1 + ln ∣ x ∣ + 1

= x^{2} + ln x^{2} + 1 + ln ∣ x ∣ + 1

Füge eine Konstante zur Lösung hinzu

= x^{2} + ln x^{2} + 1 + ln ∣ x ∣ + 1 + C

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

\int (2 x + 1)^{2} - a^{2} d x

\int 2 x csc^{2} (6 x^{2}) d x

\int \frac{x + 6}{( x + 3 ) ( x + 5 )} d x

\int + \frac{8 x - 3}{12 x ^{2} - 7 x + 1} d x