What is the integral of (x+1)/(5x^2+10x+30) ?

The integral of (x+1)/(5x^2+10x+30) is 1/10 ln|5x^2+10x+30|+C

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integral of (x+1)/(5x^2+10x+30)

\int \frac{x + 1}{5 x ^{2} + 10 x + 30} d x

\frac{1}{10} ln 5 x^{2} + 10 x + 30 + C

Solution steps

\int \frac{x + 1}{5 x ^{2} + 10 x + 30} d x

Apply u-substitution

: \int \frac{1}{10 u} d u

= \int \frac{1}{10 u} d u

Take the constant out:

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

= \frac{1}{10} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{10} ln ∣ u ∣

Substitute back

u = 5 x^{2} + 10 x + 30

= \frac{1}{10} ln 5 x^{2} + 10 x + 30

Add a constant to the solution

= \frac{1}{10} ln 5 x^{2} + 10 x + 30 + C

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What is the integral of (x+1)/(5x^2+10x+30) ?
The integral of (x+1)/(5x^2+10x+30) is 1/10 ln|5x^2+10x+30|+C