What is the integral of 1/(x^2+2x+5) ?

The integral of 1/(x^2+2x+5) is 1/2 arctan((x+1)/2)+C

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integral of 1/(x^2+2x+5)

\int \frac{1}{x ^{2} + 2 x + 5} d x

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

Solution steps

\int \frac{1}{x ^{2} + 2 x + 5} d x

Complete the square

x^{2} + 2 x + 5 : (x + 1)^{2} + 4

= \int \frac{1}{( x + 1 ) ^{2} + 4} d x

Apply u-substitution

: \int \frac{1}{u ^{2} + 4} d u

= \int \frac{1}{u ^{2} + 4} d u

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{2} arctan (v)

Substitute back

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (ln (x^{6} + 1) + y^{2})

\frac{d}{d x} (\frac{5}{12} (3 + 5) x^{3})

x \to - \infty lim (x^{0})

\frac{\partial}{\partial x} (e^{y} - x e^{x})

\frac{d}{d x} ((4 x^{2} - 3 x) e^{5 x})

What is the integral of 1/(x^2+2x+5) ?
The integral of 1/(x^2+2x+5) is 1/2 arctan((x+1)/2)+C