What is the integral of 1/(x^2+2x+26) ?

The integral of 1/(x^2+2x+26) is 1/5 arctan((x+1)/5)+C

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integral of 1/(x^2+2x+26)

\int \frac{1}{x ^{2} + 2 x + 26} d x

\frac{1}{5} arctan (\frac{x + 1}{5}) + C

Solution steps

\int \frac{1}{x ^{2} + 2 x + 26} d x

Complete the square

x^{2} + 2 x + 26 : (x + 1)^{2} + 25

= \int \frac{1}{( x + 1 ) ^{2} + 25} d x

Apply u-substitution

: \int \frac{1}{u ^{2} + 25} d u

= \int \frac{1}{u ^{2} + 25} d u

Apply Integral Substitution

= \int \frac{1}{5 ( 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}{5} \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}{5} arctan (v)

Substitute back

= \frac{1}{5} arctan (\frac{x + 1}{5})

Add a constant to the solution

= \frac{1}{5} arctan (\frac{x + 1}{5}) + C

(- 12, 0), (- 15, - 3)

\frac{d y}{d x} = - 2 x y

3 x^{\frac{2}{3}} - 2 x

\frac{s}{s + 1 0 ^{4}}

\int 6 x^{\frac{3}{5}} d x

What is the integral of 1/(x^2+2x+26) ?
The integral of 1/(x^2+2x+26) is 1/5 arctan((x+1)/5)+C