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

Lösung

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

\frac{3}{5} 5 x^{2} + 1 + \frac{1}{5} ln 5 x + 1 + 5 x^{2} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

x \to - 1 lim (- 2)

d er i v a t i v e f (x) = 2 x^{2} + 3 x - 9

\int \frac{ln ( x ^{8} )}{x} d x

d er i v a t i v e \frac{1}{3 x - 1}

\int \frac{10 x}{2 + 5 x ^{2}} d x