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Beliebt Rechnen >

integral of 1/((x^2+21)^3)

Lösung

\int \frac{1}{( x ^{2} + 21 ) ^{3}} d x

Lösung

\frac{x ^{4} arctan ( \frac{x}{21} ) + 21 x ^{3} + 42 x ^{2} arctan ( \frac{x}{21} ) + 35 21 x + 441 arctan ( \frac{x}{21} )}{1176 21 ( x ^{2} + 21 ) ^{2}} + C

Schritte zur Lösung

\int \frac{1}{( x ^{2} + 21 ) ^{3}} d x

Wende Exponentenregel an:

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

\frac{1}{( x ^{2} + 21 ) ^{3}} = (x^{2} + 21)^{- 3}

= \int (x^{2} + 21)^{- 3} d x

Wende integrale Substitution an

= \int \frac{21}{( 21 u ^{2} + 21 ) ^{3}} d u

Entferne die Konstante:

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

= 21 \cdot \int \frac{1}{( 21 u ^{2} + 21 ) ^{3}} d u

Faktorisiere

\frac{1}{( 21 u ^{2} + 21 ) ^{3}}

= 21 \cdot \int \frac{1}{9261 ( u ^{2} + 1 ) ^{3}} d u

Entferne die Konstante:

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

= 21 \frac{1}{9261} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{3}} d u

Wende integrale Reduktion an

= 21 \frac{1}{9261} (\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}})

= 21 \frac{1}{9261} (\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}{21}

ein

= 21 \frac{1}{9261} \frac{3}{4} \cdot \frac{1}{2} arctan (\frac{x}{21}) + \frac{\frac{x}{21}}{1 + ( \frac{x}{21} ) ^{2}} + \frac{\frac{x}{21}}{2 \cdot 2 ( ( \frac{x}{21} ) ^{2} + 1 ) ^{2}}

Vereinfache

21 \frac{1}{9261} \frac{3}{4} \cdot \frac{1}{2} arctan (\frac{x}{21}) + \frac{\frac{x}{21}}{1 + ( \frac{x}{21} ) ^{2}} + \frac{\frac{x}{21}}{2 \cdot 2 ( ( \frac{x}{21} ) ^{2} + 1 ) ^{2}} : \frac{x ^{4} arctan ( \frac{x}{21} ) + 21 x ^{3} + 42 x ^{2} arctan ( \frac{x}{21} ) + 35 21 x + 441 arctan ( \frac{x}{21} )}{1176 21 ( x ^{2} + 21 ) ^{2}}

= \frac{x ^{4} arctan ( \frac{x}{21} ) + 21 x ^{3} + 42 x ^{2} arctan ( \frac{x}{21} ) + 35 21 x + 441 arctan ( \frac{x}{21} )}{1176 21 ( x ^{2} + 21 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{4} arctan ( \frac{x}{21} ) + 21 x ^{3} + 42 x ^{2} arctan ( \frac{x}{21} ) + 35 21 x + 441 arctan ( \frac{x}{21} )}{1176 21 ( x ^{2} + 21 ) ^{2}} + C

Graph

Beliebte Beispiele

integral of 3/(x+1)

\int \frac{3}{x + 1} d x

xln(x)=y(1+sqrt(24+y^2))y^'(x),y(1)=1

x ln (x) = y (1 + 24 + y^{2}) y^{^{'}} (x), y (1) = 1

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sum from n=0 to infinity of (1/2)^n

n = 0 \sum \infty (\frac{1}{2})^{n}