integral of (11x^2)/((144+x^2)^2)

Soluzione

\int \frac{11 x ^{2}}{( 144 + x ^{2} ) ^{2}} d x

\frac{11 ( x ^{2} arctan ( \frac{x}{12} ) - 12 x + 144 arctan ( \frac{x}{12} ) )}{24 ( 144 + x ^{2} )} + C

Fasi della soluzione

\int \frac{11 x ^{2}}{( 144 + x ^{2} ) ^{2}} d x

Porta fuori la costante:

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

= 11 \cdot \int \frac{x ^{2}}{( 144 + x ^{2} ) ^{2}} d x

Applicare la sostituzione integrale

= 11 \cdot \int \frac{u ^{2}}{12 ( u ^{2} + 1 ) ^{2}} d u

Porta fuori la costante:

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

= 11 \cdot \frac{1}{12} \cdot \int \frac{u ^{2}}{( u ^{2} + 1 ) ^{2}} d u

Prendi la frazione parziale di

\frac{u ^{2}}{( u ^{2} + 1 ) ^{2}} : \frac{1}{u ^{2} + 1} - \frac{1}{( u ^{2} + 1 ) ^{2}}

= 11 \cdot \frac{1}{12} \cdot \int \frac{1}{u ^{2} + 1} - \frac{1}{( u ^{2} + 1 ) ^{2}} d u

Applica la Regola della Somma:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 11 \cdot \frac{1}{12} (\int \frac{1}{u ^{2} + 1} d u - \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u)

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

\int \frac{1}{( u ^{2} + 1 ) ^{2}} d u = \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}})

= 11 \cdot \frac{1}{12} (arctan (u) - \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

Sostituire indietro

u = \frac{x}{12}

= 11 \cdot \frac{1}{12} (arctan (\frac{x}{12}) - \frac{1}{2} (arctan (\frac{x}{12}) + \frac{\frac{x}{12}}{1 + ( \frac{x}{12} ) ^{2}}))

Semplificare

11 \cdot \frac{1}{12} (arctan (\frac{x}{12}) - \frac{1}{2} (arctan (\frac{x}{12}) + \frac{\frac{x}{12}}{1 + ( \frac{x}{12} ) ^{2}})) : \frac{11 ( x ^{2} arctan ( \frac{x}{12} ) - 12 x + 144 arctan ( \frac{x}{12} ) )}{24 ( 144 + x ^{2} )}

= \frac{11 ( x ^{2} arctan ( \frac{x}{12} ) - 12 x + 144 arctan ( \frac{x}{12} ) )}{24 ( 144 + x ^{2} )}

Aggiungi una costante alla soluzione

= \frac{11 ( x ^{2} arctan ( \frac{x}{12} ) - 12 x + 144 arctan ( \frac{x}{12} ) )}{24 ( 144 + x ^{2} )} + C

\frac{d}{d x} (x^{6} ln (7 x))

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

\frac{d}{d x} (x^{\frac{9}{4}})

f (θ) = 11 θ

f (s) = \frac{s ^{2} + 1}{s ^{2} + 4}