What is the integral of 4/(5+2x+x^2) ?

The integral of 4/(5+2x+x^2) is 2arctan(1/2 (x+1))+C

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integral of 4/(5+2x+x^2)

\int \frac{4}{5 + 2 x + x ^{2}} d x

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

Solution steps

\int \frac{4}{5 + 2 x + x ^{2}} d x

Take the constant out:

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

= 4 \cdot \int \frac{1}{5 + 2 x + x ^{2}} d x

Complete the square

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

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

Apply u-substitution

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

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

Apply Integral Substitution

= 4 \cdot \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

= 4 \cdot \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)

= 4 \cdot \frac{1}{2} arctan (v)

Substitute back

= 4 \cdot \frac{1}{2} arctan (\frac{x + 1}{2})

Simplify

4 \cdot \frac{1}{2} arctan (\frac{x + 1}{2}) : 2 arctan (\frac{1}{2} (x + 1))

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

Add a constant to the solution

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

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What is the integral of 4/(5+2x+x^2) ?
The integral of 4/(5+2x+x^2) is 2arctan(1/2 (x+1))+C