integral of ((x^4))/(x^2+4)

Lösung

\int \frac{( x ^{4} )}{x ^{2} + 4} d x

8 (\frac{x ^{3}}{24} - \frac{x}{2} + arctan (\frac{x}{2})) + C

Schritte zur Lösung

\int \frac{x ^{4}}{x ^{2} + 4} d x

Wende integrale Substitution an

= \int \frac{8 u ^{4}}{u ^{2} + 1} d u

Entferne die Konstante:

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

= 8 \cdot \int \frac{u ^{4}}{u ^{2} + 1} d u

Lange Division

\frac{u ^{4}}{u ^{2} + 1} : u^{2} - 1 + \frac{1}{u ^{2} + 1}

= 8 \cdot \int u^{2} - 1 + \frac{1}{u ^{2} + 1} 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

= 8 (\int u^{2} d u - \int 1 d u + \int \frac{1}{u ^{2} + 1} d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int 1 d u = u

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

= 8 (\frac{u ^{3}}{3} - u + arctan (u))

Setze in

u = \frac{x}{2}

ein

= 8 (\frac{( \frac{x}{2} ) ^{3}}{3} - \frac{x}{2} + arctan (\frac{x}{2}))

Vereinfache

8 (\frac{( \frac{x}{2} ) ^{3}}{3} - \frac{x}{2} + arctan (\frac{x}{2})) : 8 (\frac{x ^{3}}{24} - \frac{x}{2} + arctan (\frac{x}{2}))

= 8 (\frac{x ^{3}}{24} - \frac{x}{2} + arctan (\frac{x}{2}))

Füge eine Konstante zur Lösung hinzu

= 8 (\frac{x ^{3}}{24} - \frac{x}{2} + arctan (\frac{x}{2})) + C

\frac{d y}{d x} = \frac{- y ^{2}}{1 + y ^{2}} e^{- x}, y (0) = 1

x \to - 2 lim (h (x))

d er i v a t i v e e^{- 7 x}

\int x^{2} 2 x + 1 d x

x \to 0 - lim (x + 1)