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

Lösung

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

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

Wende Exponentenregel an:

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

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

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

\int \frac{e}{x} d x

a re a y = 5 - x^{2}, y = x^{2} + 3

\int \frac{cos ( x )}{( 1 + sin ( x ) )} d x

d er i v a t i v e f (x) = 2 x^{3} - 3 x^{2} - 36 x + 5

l a pl a ce t r an s f or m f (x) = cos (a x)