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

Lösung

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

- \frac{1}{2} ln 1 - 4 x^{2} + 1 + \frac{1}{2} ln 1 - 4 x^{2} - 1 + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{1}{u ^{2} - 1} d u

Faktorisiere

u^{2} - 1 : - (- u^{2} + 1)

= \int \frac{1}{- ( - u ^{2} + 1 )} d u

Entferne die Konstante:

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

= - \int \frac{1}{- u ^{2} + 1} d u

Nutze das gemeinsame Integral :

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

= - (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2})

Setze in

u = 1 - 4 x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln 1 - 4 x^{2} + 1 + \frac{1}{2} ln 1 - 4 x^{2} - 1 + C

d er i v a t i v e f (x) = \frac{x ^{4}}{4} - 2 x^{2} - 4

d er i v a t i v e x^{2} - 2 x + 4

d er i v a t i v e f (x) = 4 x^{\frac{5}{4}} + 6 x^{\frac{3}{2}} + 3 x

y^{^{''}} + 12 y^{^{'}} + 36 y = 0, y (1) = 0, y^{^{'}} (1) = 1

d er i v a t i v e 2^{60}