integral of 1/((x^2+16)^3)

Lösung

\int \frac{1}{( x ^{2} + 16 ) ^{3}} d x

\frac{3 x ^{4} arctan ( \frac{x}{4} ) + 12 x ^{3} + 96 x ^{2} arctan ( \frac{x}{4} ) + 320 x + 768 arctan ( \frac{x}{4} )}{8192 ( x ^{2} + 16 ) ^{2}} + C

Schritte zur Lösung

\int \frac{1}{( x ^{2} + 16 ) ^{3}} d x

Wende Exponentenregel an:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{( x ^{2} + 16 ) ^{3}} = (x^{2} + 16)^{- 3}

= \int (x^{2} + 16)^{- 3} d x

Wende integrale Substitution an

= \int \frac{1}{1024 ( u ^{2} + 1 ) ^{3}} d u

Entferne die Konstante:

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

= \frac{1}{1024} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{3}} d u

Wende integrale Reduktion an

= \frac{1}{1024} (\frac{3}{4} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u + \frac{u}{2 \cdot 2 ( u ^{2} + 1 ) ^{2}})

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

= \frac{1}{1024} (\frac{3}{4} \cdot \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}) + \frac{u}{2 \cdot 2 ( u ^{2} + 1 ) ^{2}})

Setze in

u = \frac{x}{4}

ein

= \frac{1}{1024} \frac{3}{4} \cdot \frac{1}{2} (arctan (\frac{x}{4}) + \frac{\frac{x}{4}}{1 + ( \frac{x}{4} ) ^{2}}) + \frac{\frac{x}{4}}{2 \cdot 2 ( ( \frac{x}{4} ) ^{2} + 1 ) ^{2}}

Vereinfache

\frac{1}{1024} \frac{3}{4} \cdot \frac{1}{2} (arctan (\frac{x}{4}) + \frac{\frac{x}{4}}{1 + ( \frac{x}{4} ) ^{2}}) + \frac{\frac{x}{4}}{2 \cdot 2 ( ( \frac{x}{4} ) ^{2} + 1 ) ^{2}} : \frac{3 x ^{4} arctan ( \frac{x}{4} ) + 12 x ^{3} + 96 x ^{2} arctan ( \frac{x}{4} ) + 320 x + 768 arctan ( \frac{x}{4} )}{8192 ( x ^{2} + 16 ) ^{2}}

= \frac{3 x ^{4} arctan ( \frac{x}{4} ) + 12 x ^{3} + 96 x ^{2} arctan ( \frac{x}{4} ) + 320 x + 768 arctan ( \frac{x}{4} )}{8192 ( x ^{2} + 16 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{3 x ^{4} arctan ( \frac{x}{4} ) + 12 x ^{3} + 96 x ^{2} arctan ( \frac{x}{4} ) + 320 x + 768 arctan ( \frac{x}{4} )}{8192 ( x ^{2} + 16 ) ^{2}} + C

\int \frac{1}{3 x ^{2} + 12} d x

\int u^{- 5} (7 - u^{5} + u^{10}) d u

\frac{d y}{d x} - 12 y = e^{9 x} y^{2}

\frac{\partial}{\partial x} (4 x y e^{x - y^{2}})

x \to 0 lim (e^{x^{2}} - 1)