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Beliebt Rechnen >

integral of 1/((x^2+9)^3)

Lösung

\int \frac{1}{( x ^{2} + 9 ) ^{3}} d x

Lösung

\frac{x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 18 x ^{2} arctan ( \frac{x}{3} ) + 45 x + 81 arctan ( \frac{x}{3} )}{648 ( x ^{2} + 9 ) ^{2}} + C

Schritte zur Lösung

\int \frac{1}{( x ^{2} + 9 ) ^{3}} d x

Wende Exponentenregel an:

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

\frac{1}{( x ^{2} + 9 ) ^{3}} = (x^{2} + 9)^{- 3}

= \int (x^{2} + 9)^{- 3} d x

Wende integrale Substitution an

= \int \frac{1}{243 ( 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}{243} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{3}} d u

Wende integrale Reduktion an

= \frac{1}{243} (\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}{243} (\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}{3}

ein

= \frac{1}{243} \frac{3}{4} \cdot \frac{1}{2} (arctan (\frac{x}{3}) + \frac{\frac{x}{3}}{1 + ( \frac{x}{3} ) ^{2}}) + \frac{\frac{x}{3}}{2 \cdot 2 ( ( \frac{x}{3} ) ^{2} + 1 ) ^{2}}

Vereinfache

\frac{1}{243} \frac{3}{4} \cdot \frac{1}{2} (arctan (\frac{x}{3}) + \frac{\frac{x}{3}}{1 + ( \frac{x}{3} ) ^{2}}) + \frac{\frac{x}{3}}{2 \cdot 2 ( ( \frac{x}{3} ) ^{2} + 1 ) ^{2}} : \frac{x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 18 x ^{2} arctan ( \frac{x}{3} ) + 45 x + 81 arctan ( \frac{x}{3} )}{648 ( x ^{2} + 9 ) ^{2}}

= \frac{x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 18 x ^{2} arctan ( \frac{x}{3} ) + 45 x + 81 arctan ( \frac{x}{3} )}{648 ( x ^{2} + 9 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 18 x ^{2} arctan ( \frac{x}{3} ) + 45 x + 81 arctan ( \frac{x}{3} )}{648 ( x ^{2} + 9 ) ^{2}} + C

Graph

Beliebte Beispiele

derivative (x^4+5x^2-5)^4

d er i v a t i v e (x^{4} + 5 x^{2} - 5)^{4}

(\partial)/(\partial x)(2+xln(xy-9))

\frac{\partial}{\partial x} (2 + x ln (x y - 9))

laplacetransform u^1

l a pl a ce t r an s f or m u^{1}

derivative f(x)= 3/8 x^4-2-(10)/x

d er i v a t i v e f (x) = \frac{3}{8} x^{4} - 2 - \frac{10}{x}

derivative f(x)=(3x^2-1)^2

d er i v a t i v e f (x) = (3 x^{2} - 1)^{2}