What is the integral of (3x+6)/(x^2+2x+2) ?

The integral of (3x+6)/(x^2+2x+2) is 3/2 ln|x^2+2x+2|+3arctan(x+1)+C

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integral of (3x+6)/(x^2+2x+2)

\int \frac{3 x + 6}{x ^{2} + 2 x + 2} d x

\frac{3}{2} ln x^{2} + 2 x + 2 + 3 arctan (x + 1) + C

Solution steps

\int \frac{3 x + 6}{x ^{2} + 2 x + 2} d x

Expand

\frac{3 x + 6}{x ^{2} + 2 x + 2} : \frac{3 x}{x ^{2} + 2 x + 2} + \frac{6}{x ^{2} + 2 x + 2}

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{3 x}{x ^{2} + 2 x + 2} d x + \int \frac{6}{x ^{2} + 2 x + 2} d x

\int \frac{3 x}{x ^{2} + 2 x + 2} d x = 3 (\frac{1}{2} ln x^{2} + 2 x + 2 - arctan (x + 1))

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

= 3 (\frac{1}{2} ln x^{2} + 2 x + 2 - arctan (x + 1)) + 6 arctan (x + 1)

Simplify

3 (\frac{1}{2} ln x^{2} + 2 x + 2 - arctan (x + 1)) + 6 arctan (x + 1) : \frac{3}{2} ln x^{2} + 2 x + 2 + 3 arctan (x + 1)

= \frac{3}{2} ln x^{2} + 2 x + 2 + 3 arctan (x + 1)

Add a constant to the solution

= \frac{3}{2} ln x^{2} + 2 x + 2 + 3 arctan (x + 1) + C

\int_{0}^{4} x + 3 x d x

\int \frac{x ^{6} - x ^{3} + 64}{x ^{4} + 8 x ^{2}} d x

\frac{d}{d x} (1 - ln (1 - x))

\int \frac{1}{8 y ^{2}} d y

\frac{d}{d x} (6 x y - 12)

What is the integral of (3x+6)/(x^2+2x+2) ?
The integral of (3x+6)/(x^2+2x+2) is 3/2 ln|x^2+2x+2|+3arctan(x+1)+C