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Beliebt Rechnen >

integral of 1/((9x^2+4)^3)

Lösung

\int \frac{1}{( 9 x ^{2} + 4 ) ^{3}} d x

Lösung

\frac{81 x ^{4} arctan ( \frac{3 x}{2} ) + 54 x ^{3} + 72 x ^{2} arctan ( \frac{3 x}{2} ) + 40 x + 16 arctan ( \frac{3 x}{2} )}{256 ( 9 x ^{2} + 4 ) ^{2}} + C

Schritte zur Lösung

\int \frac{1}{( 9 x ^{2} + 4 ) ^{3}} d x

Wende integrale Substitution an

= \int \frac{1}{96 ( 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}{96} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{3}} d u

Wende integrale Reduktion an

= \frac{1}{96} (\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}{96} (\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{3}{2} x

ein

= \frac{1}{96} \frac{3}{4} \cdot \frac{1}{2} (arctan (\frac{3}{2} x) + \frac{\frac{3}{2} x}{1 + ( \frac{3}{2} x ) ^{2}}) + \frac{\frac{3}{2} x}{2 \cdot 2 ( ( \frac{3}{2} x ) ^{2} + 1 ) ^{2}}

Vereinfache

\frac{1}{96} \frac{3}{4} \cdot \frac{1}{2} (arctan (\frac{3}{2} x) + \frac{\frac{3}{2} x}{1 + ( \frac{3}{2} x ) ^{2}}) + \frac{\frac{3}{2} x}{2 \cdot 2 ( ( \frac{3}{2} x ) ^{2} + 1 ) ^{2}} : \frac{81 x ^{4} arctan ( \frac{3 x}{2} ) + 54 x ^{3} + 72 x ^{2} arctan ( \frac{3 x}{2} ) + 40 x + 16 arctan ( \frac{3 x}{2} )}{256 ( 9 x ^{2} + 4 ) ^{2}}

= \frac{81 x ^{4} arctan ( \frac{3 x}{2} ) + 54 x ^{3} + 72 x ^{2} arctan ( \frac{3 x}{2} ) + 40 x + 16 arctan ( \frac{3 x}{2} )}{256 ( 9 x ^{2} + 4 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{81 x ^{4} arctan ( \frac{3 x}{2} ) + 54 x ^{3} + 72 x ^{2} arctan ( \frac{3 x}{2} ) + 40 x + 16 arctan ( \frac{3 x}{2} )}{256 ( 9 x ^{2} + 4 ) ^{2}} + C

Graph

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