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

Lösung

\int \frac{x ^{2}}{1 - 4 x ^{2}} d x

\frac{x ^{3}}{1 - 4 x ^{2}} - 2 (- \frac{1}{32} ln ∣ 2 x + 1 ∣ - \frac{1}{32 ( 2 x + 1 )} + \frac{1}{32} ln ∣ 2 x - 1 ∣ - \frac{1}{32 ( 2 x - 1 )}) + C

Schritte zur Lösung

\int \frac{x ^{2}}{1 - 4 x ^{2}} d x

Wende die partielle Integration an

= \frac{x ^{3}}{1 - 4 x ^{2}} - \int \frac{2 x ^{2}}{( 1 - 4 x ^{2} ) ^{2}} d x

\int \frac{2 x ^{2}}{( 1 - 4 x ^{2} ) ^{2}} d x = 2 (- \frac{1}{32} ln ∣ 2 x + 1 ∣ - \frac{1}{32 ( 2 x + 1 )} + \frac{1}{32} ln ∣ 2 x - 1 ∣ - \frac{1}{32 ( 2 x - 1 )})

= \frac{x ^{3}}{1 - 4 x ^{2}} - 2 (- \frac{1}{32} ln ∣ 2 x + 1 ∣ - \frac{1}{32 ( 2 x + 1 )} + \frac{1}{32} ln ∣ 2 x - 1 ∣ - \frac{1}{32 ( 2 x - 1 )})

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{1 - 4 x ^{2}} - 2 (- \frac{1}{32} ln ∣ 2 x + 1 ∣ - \frac{1}{32 ( 2 x + 1 )} + \frac{1}{32} ln ∣ 2 x - 1 ∣ - \frac{1}{32 ( 2 x - 1 )}) + C

x \to \infty lim (4 \cdot x)

x \to + 0 + lim (e^{- \frac{2}{x ^{3}}})

d er i v a t i v e f (x) = \frac{x ^{2}}{cos ( x )}

d er i v a t i v e f (x) = tan (5 x)

\int sec^{- \frac{1}{2}} (x) tan^{3} (x) d x