What is the integral of (x+2)/(sqrt(x^2+6x+36)) ?

The integral of (x+2)/(sqrt(x^2+6x+36)) is sqrt(x^2+6x+36)-ln|x+sqrt(x^2+6x+36)+3|+3/2 ln(3)+C

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integral of (x+2)/(sqrt(x^2+6x+36))

\int \frac{x + 2}{x ^{2} + 6 x + 36} d x

x^{2} + 6 x + 36 - ln x + x^{2} + 6 x + 36 + 3 + \frac{3}{2} ln (3) + C

Solution steps

\int \frac{x + 2}{x ^{2} + 6 x + 36} d x

Expand

\frac{x + 2}{x ^{2} + 6 x + 36} : \frac{x}{x ^{2} + 6 x + 36} + \frac{2}{x ^{2} + 6 x + 36}

= \int \frac{x}{x ^{2} + 6 x + 36} + \frac{2}{x ^{2} + 6 x + 36} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x}{x ^{2} + 6 x + 36} d x + \int \frac{2}{x ^{2} + 6 x + 36} d x

\int \frac{x}{x ^{2} + 6 x + 36} d x = x^{2} + 6 x + 36 - 3 (ln x + 3 + x^{2} + 6 x + 36 - \frac{3}{2} ln (3))

\int \frac{2}{x ^{2} + 6 x + 36} d x = 2 ln x + 3 + x^{2} + 6 x + 36 - 3 ln (3)

= x^{2} + 6 x + 36 - 3 (ln x + 3 + x^{2} + 6 x + 36 - \frac{3}{2} ln (3)) + 2 ln x + 3 + x^{2} + 6 x + 36 - 3 ln (3)

Simplify

x^{2} + 6 x + 36 - 3 (ln x + 3 + x^{2} + 6 x + 36 - \frac{3}{2} ln (3)) + 2 ln x + 3 + x^{2} + 6 x + 36 - 3 ln (3) : x^{2} + 6 x + 36 - ln x + x^{2} + 6 x + 36 + 3 + \frac{3}{2} ln (3)

= x^{2} + 6 x + 36 - ln x + x^{2} + 6 x + 36 + 3 + \frac{3}{2} ln (3)

Add a constant to the solution

= x^{2} + 6 x + 36 - ln x + x^{2} + 6 x + 36 + 3 + \frac{3}{2} ln (3) + C

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What is the integral of (x+2)/(sqrt(x^2+6x+36)) ?
The integral of (x+2)/(sqrt(x^2+6x+36)) is sqrt(x^2+6x+36)-ln|x+sqrt(x^2+6x+36)+3|+3/2 ln(3)+C