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

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

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

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

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

Solution steps

\int \frac{x + 2}{x ^{2} + 2 x + 2} d x

Complete the square

x^{2} + 2 x + 2 : (x + 1)^{2} + 1

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

Apply u-substitution

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

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

Expand

\frac{u + 1}{u ^{2} + 1} : \frac{u}{u ^{2} + 1} + \frac{1}{u ^{2} + 1}

Apply the Sum Rule:

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

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

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

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

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

Substitute back

u = x + 1

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

Simplify

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

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

Add a constant to the solution

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

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