integral of (x^3)/(a^2-x^2)

Lösung

\int \frac{x ^{3}}{a ^{2} - x ^{2}} d x

- \frac{1}{2} (- a^{2} + x^{2} + a^{2} ln a^{2} - x^{2}) + C

Schritte zur Lösung

\int \frac{x ^{3}}{a ^{2} - x ^{2}} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Multipliziere aus

\frac{- u + a ^{2}}{u} : - 1 + \frac{a ^{2}}{u}

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d u = u

\int \frac{a ^{2}}{u} d u = a^{2} ln ∣ u ∣

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

Setze in

u = a^{2} - x^{2}

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e \frac{t ^{5} - 3 t ^{2}}{2 t}

\int \frac{x}{x + 5} d x

d er i v a t i v e f (x) = 7 e^{- x} - 4 e^{- 7 x}

\frac{\partial}{\partial x} (3 sec^{2} (3 x - 2 y + 1))

\frac{d}{d x} ((2 x + 3)^{5} + 10 x (2 x + 3)^{4})