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

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

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

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

x^{2} + 2 x + 2 + 2 ln x^{2} + 2 x + 2 + x + 1 + C

Solution steps

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

Expand

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

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

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

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

\int \frac{3}{x ^{2} + 2 x + 2} d x = 3 ln x^{2} + 2 x + 2 + x + 1

= x^{2} + 2 x + 2 - ln x^{2} + 2 x + 2 + x + 1 + 3 ln x^{2} + 2 x + 2 + x + 1

Simplify

= x^{2} + 2 x + 2 + 2 ln x^{2} + 2 x + 2 + x + 1

Add a constant to the solution

= x^{2} + 2 x + 2 + 2 ln x^{2} + 2 x + 2 + x + 1 + C

x^{- 4}

\int \frac{tan ( 3 x ) - cot ( 3 x )}{sin ( 3 x )} d x

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

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