What is the integral of 1/((x+1)(x^2+2x+2)^2) ?

The integral of 1/((x+1)(x^2+2x+2)^2) is ln(x+1)-1/2 ln|x^2+2x+2|+1/(2(x^2+2x+2))+C

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

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

ln (x + 1) - \frac{1}{2} ln x^{2} + 2 x + 2 + \frac{1}{2 ( x ^{2} + 2 x + 2 )} + C

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

: \int \frac{1}{2 ( v - 1 ) v ^{2}} d v

= \int \frac{1}{2 ( v - 1 ) v ^{2}} d v

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{( v - 1 ) v ^{2}} d v

Take the partial fraction of

\frac{1}{( v - 1 ) v ^{2}} : \frac{1}{v - 1} - \frac{1}{v} - \frac{1}{v ^{2}}

= \frac{1}{2} \cdot \int \frac{1}{v - 1} - \frac{1}{v} - \frac{1}{v ^{2}} d v

Apply the Sum Rule:

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

= \frac{1}{2} (\int \frac{1}{v - 1} d v - \int \frac{1}{v} d v - \int \frac{1}{v ^{2}} d v)

\int \frac{1}{v - 1} d v = ln ∣ v - 1 ∣

\int \frac{1}{v} d v = ln ∣ v ∣

\int \frac{1}{v ^{2}} d v = - \frac{1}{v}

= \frac{1}{2} (ln ∣ v - 1 ∣ - ln ∣ v ∣ - (- \frac{1}{v}))

Substitute back

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

Simplify

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

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

Add a constant to the solution

= ln (x + 1) - \frac{1}{2} ln x^{2} + 2 x + 2 + \frac{1}{2 ( x ^{2} + 2 x + 2 )} + C

ln (x e^{- 3 x})

f (x) = \frac{9}{x}

\int e^{- x^{2} \frac{1}{2}} d x

\frac{d}{d x} (\frac{2 x + 6 x ^{2}}{( 1 + 6 x ) ^{2}})

y + x^{2} + y^{2} d x - x d y = 0

What is the integral of 1/((x+1)(x^2+2x+2)^2) ?
The integral of 1/((x+1)(x^2+2x+2)^2) is ln(x+1)-1/2 ln|x^2+2x+2|+1/(2(x^2+2x+2))+C