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

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

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

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

2 arctan (x + 1) + C

Solution steps

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

Take the constant out:

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

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

Complete the square

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

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

Apply u-substitution

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

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

Use the common integral:

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

= 2 arctan (u)

Substitute back

u = x + 1

= 2 arctan (x + 1)

Add a constant to the solution

= 2 arctan (x + 1) + C

x \to 1 - lim (x^{2} - 1)

x \to 1 lim (\frac{x - 1}{x ^{2} - 1})

\frac{d}{d x} (x^{3} (ln (x))^{c o s (x)})

\int y^{- 4} d y

f (x) = \frac{3}{( x + 5 ) ^{2}}, at x = \frac{3}{16}

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