What is the integral of (x+1)/(9x^2+18x+10) ?

The integral of (x+1)/(9x^2+18x+10) is 1/18 ln|9x^2+18x+10|+C

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integral of (x+1)/(9x^2+18x+10)

\int \frac{x + 1}{9 x ^{2} + 18 x + 10} d x

\frac{1}{18} ln 9 x^{2} + 18 x + 10 + C

Solution steps

\int \frac{x + 1}{9 x ^{2} + 18 x + 10} d x

Apply u-substitution

: \int \frac{1}{18 u} d u

= \int \frac{1}{18 u} d u

Take the constant out:

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

= \frac{1}{18} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{18} ln ∣ u ∣

Substitute back

u = 9 x^{2} + 18 x + 10

= \frac{1}{18} ln 9 x^{2} + 18 x + 10

Add a constant to the solution

= \frac{1}{18} ln 9 x^{2} + 18 x + 10 + C

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What is the integral of (x+1)/(9x^2+18x+10) ?
The integral of (x+1)/(9x^2+18x+10) is 1/18 ln|9x^2+18x+10|+C