What is the integral of 2/(10+6x+x^2) ?

The integral of 2/(10+6x+x^2) is 2arctan(x+3)+C

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integral of 2/(10+6x+x^2)

\int \frac{2}{10 + 6 x + x ^{2}} d x

2 arctan (x + 3) + C

Solution steps

\int \frac{2}{10 + 6 x + 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}{10 + 6 x + x ^{2}} d x

Complete the square

10 + 6 x + x^{2} : (x + 3)^{2} + 1

= 2 \cdot \int \frac{1}{( x + 3 ) ^{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 + 3

= 2 arctan (x + 3)

Add a constant to the solution

= 2 arctan (x + 3) + C

\frac{1}{4 x}

\int (1 - x) (1 - 2 x) (1 - 3 x) d x

\int \frac{sin ^{2} ( x ) cos ( 2 x )}{2} d x

\int_{1}^{2} \frac{( 2 + 2 x - x ^{2} )}{8} d x

x^{^{''}} - 12 x^{^{'}} + 36 x = 2 t e^{6 t}

What is the integral of 2/(10+6x+x^2) ?
The integral of 2/(10+6x+x^2) is 2arctan(x+3)+C