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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

\int 20 - 3 16 - 2 x d x

y = 3 x, y = x^{2}

e^{x} ln (1 + x)

x \to 4 lim (\frac{3 - x}{x - 4})

\frac{1}{( s + 1 ) ( s + 2 )}

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