What is the integral of (13)/(4x^2+12x+9) ?

The integral of (13)/(4x^2+12x+9) is -(13)/(2(2x+3))+C

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integral of (13)/(4x^2+12x+9)

\int \frac{13}{4 x ^{2} + 12 x + 9} d x

- \frac{13}{2 ( 2 x + 3 )} + C

Solution steps

\int \frac{13}{4 x ^{2} + 12 x + 9} d x

Take the constant out:

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

= 13 \cdot \int \frac{1}{4 x ^{2} + 12 x + 9} d x

Complete the square

4 x^{2} + 12 x + 9 : 4 (x + \frac{3}{2})^{2}

= 13 \cdot \int \frac{1}{4 ( x + \frac{3}{2} ) ^{2}} d x

Apply u-substitution

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

= 13 \cdot \int \frac{1}{4 u ^{2}} d u

Take the constant out:

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

= 13 \cdot \frac{1}{4} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= 13 \cdot \frac{1}{4} (- \frac{1}{u})

Substitute back

u = x + \frac{3}{2}

= 13 \cdot \frac{1}{4} (- \frac{1}{x + \frac{3}{2}})

Simplify

13 \cdot \frac{1}{4} (- \frac{1}{x + \frac{3}{2}}) : - \frac{13}{2 ( 2 x + 3 )}

= - \frac{13}{2 ( 2 x + 3 )}

Add a constant to the solution

= - \frac{13}{2 ( 2 x + 3 )} + C

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What is the integral of (13)/(4x^2+12x+9) ?
The integral of (13)/(4x^2+12x+9) is -(13)/(2(2x+3))+C