integral of 1/((x^2+25)^3)

Lösung

\int \frac{1}{( x ^{2} + 25 ) ^{3}} d x

\frac{3 x ^{4} arctan ( \frac{x}{5} ) + 15 x ^{3} + 150 x ^{2} arctan ( \frac{x}{5} ) + 625 x + 1875 arctan ( \frac{x}{5} )}{25000 ( x ^{2} + 25 ) ^{2}} + C

Schritte zur Lösung

\int \frac{1}{( x ^{2} + 25 ) ^{3}} d x

Wende Exponentenregel an:

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

\frac{1}{( x ^{2} + 25 ) ^{3}} = (x^{2} + 25)^{- 3}

= \int (x^{2} + 25)^{- 3} d x

Wende integrale Substitution an

= \int \frac{1}{3125 ( 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}{3125} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{3}} d u

Wende integrale Reduktion an

= \frac{1}{3125} (\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}{3125} (\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}{5}

ein

= \frac{1}{3125} \frac{3}{4} \cdot \frac{1}{2} (arctan (\frac{x}{5}) + \frac{\frac{x}{5}}{1 + ( \frac{x}{5} ) ^{2}}) + \frac{\frac{x}{5}}{2 \cdot 2 ( ( \frac{x}{5} ) ^{2} + 1 ) ^{2}}

Vereinfache

\frac{1}{3125} \frac{3}{4} \cdot \frac{1}{2} (arctan (\frac{x}{5}) + \frac{\frac{x}{5}}{1 + ( \frac{x}{5} ) ^{2}}) + \frac{\frac{x}{5}}{2 \cdot 2 ( ( \frac{x}{5} ) ^{2} + 1 ) ^{2}} : \frac{3 x ^{4} arctan ( \frac{x}{5} ) + 15 x ^{3} + 150 x ^{2} arctan ( \frac{x}{5} ) + 625 x + 1875 arctan ( \frac{x}{5} )}{25000 ( x ^{2} + 25 ) ^{2}}

= \frac{3 x ^{4} arctan ( \frac{x}{5} ) + 15 x ^{3} + 150 x ^{2} arctan ( \frac{x}{5} ) + 625 x + 1875 arctan ( \frac{x}{5} )}{25000 ( x ^{2} + 25 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{3 x ^{4} arctan ( \frac{x}{5} ) + 15 x ^{3} + 150 x ^{2} arctan ( \frac{x}{5} ) + 625 x + 1875 arctan ( \frac{x}{5} )}{25000 ( x ^{2} + 25 ) ^{2}} + C

\frac{\partial}{\partial x} ((3 x^{2} + 5 x + 1)^{\frac{3}{2}})

\int x \cdot \frac{5}{2} \cdot x^{2} + 7 d x

\int_{0}^{2} 2^{- x} d x

\frac{\partial}{\partial y} (\frac{1}{x ^{2} + y})

x \to 0 lim (2 (1 + 3 x)^{\frac{2}{x}})