What is the integral of 1/(x(x^{999)+1)^2} ?

The integral of 1/(x(x^{999)+1)^2} is 1/999 (ln|x^{999}|-ln|x^{999}+1|+1/(x^{999)+1})+C

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integral of 1/(x(x^{999)+1)^2}

\int \frac{1}{x ( x ^{999} + 1 ) ^{2}} d x

\frac{1}{999} (ln x^{999} - ln x^{999} + 1 + \frac{1}{x ^{999} + 1}) + C

Solution steps

\int \frac{1}{x ( x ^{999} + 1 ) ^{2}} d x

Apply u-substitution

: \int \frac{1}{999 u ( u + 1 ) ^{2}} d u

= \int \frac{1}{999 u ( u + 1 ) ^{2}} d u

Take the constant out:

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

= \frac{1}{999} \cdot \int \frac{1}{u ( u + 1 ) ^{2}} d u

Apply u-substitution

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

= \frac{1}{999} \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}{999} \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}{999} (\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}{999} (ln ∣ v - 1 ∣ - ln ∣ v ∣ - (- \frac{1}{v}))

Substitute back

= \frac{1}{999} (ln x^{999} + 1 - 1 - ln x^{999} + 1 - (- \frac{1}{x ^{999} + 1}))

Simplify

= \frac{1}{999} (ln x^{999} - ln x^{999} + 1 + \frac{1}{x ^{999} + 1})

Add a constant to the solution

= \frac{1}{999} (ln x^{999} - ln x^{999} + 1 + \frac{1}{x ^{999} + 1}) + C

f (x) = 2 x^{3}, (1, 2)

\frac{d}{d y} (ln (\frac{( 2 y + 1 ) ^{4}}{y ^{2} + 1}))

\frac{2}{x - 5}

\int \frac{- 1400 x}{25 - x ^{2}} d x

\frac{d}{d x} (sin (x) cos (x) + sin (x))

What is the integral of 1/(x(x^{999)+1)^2} ?
The integral of 1/(x(x^{999)+1)^2} is 1/999 (ln|x^{999}|-ln|x^{999}+1|+1/(x^{999)+1})+C