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

The integral of 1/((x+2)^2+9) is 1/3 arctan((x+2)/3)+C

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

\int \frac{1}{( x + 2 ) ^{2} + 9} d x

\frac{1}{3} arctan (\frac{x + 2}{3}) + C

Solution steps

\int \frac{1}{( x + 2 ) ^{2} + 9} d x

Apply u-substitution

: \int \frac{1}{u ^{2} + 9} d u

= \int \frac{1}{u ^{2} + 9} d u

Apply Integral Substitution

= \int \frac{1}{3 ( v ^{2} + 1 )} d v

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= \frac{1}{3} arctan (v)

Substitute back

= \frac{1}{3} arctan (\frac{x + 2}{3})

Add a constant to the solution

= \frac{1}{3} arctan (\frac{x + 2}{3}) + C

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What is the integral of 1/((x+2)^2+9) ?
The integral of 1/((x+2)^2+9) is 1/3 arctan((x+2)/3)+C