What is the integral of 9/(x^2-6x+18) ?

The integral of 9/(x^2-6x+18) is 3arctan(1/3 (x-3))+C

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integral of 9/(x^2-6x+18)

\int \frac{9}{x ^{2} - 6 x + 18} d x

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

Solution steps

\int \frac{9}{x ^{2} - 6 x + 18} d x

Take the constant out:

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

= 9 \cdot \int \frac{1}{x ^{2} - 6 x + 18} d x

Complete the square

x^{2} - 6 x + 18 : (x - 3)^{2} + 9

= 9 \cdot \int \frac{1}{( x - 3 ) ^{2} + 9} d x

Apply u-substitution

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

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

Apply Integral Substitution

= 9 \cdot \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

= 9 \cdot \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)

= 9 \cdot \frac{1}{3} arctan (v)

Substitute back

= 9 \cdot \frac{1}{3} arctan (\frac{x - 3}{3})

Simplify

9 \cdot \frac{1}{3} arctan (\frac{x - 3}{3}) : 3 arctan (\frac{1}{3} (x - 3))

= 3 arctan (\frac{1}{3} (x - 3))

Add a constant to the solution

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

\frac{0.6}{s}

\int_{- 1}^{0} - 1 d x

\int cos (\frac{nπ x}{L}) sin (\frac{nπ x}{L}) d x

f (x) = \frac{20}{x - 1}, at x = 5

t^{2} t^{3}

What is the integral of 9/(x^2-6x+18) ?
The integral of 9/(x^2-6x+18) is 3arctan(1/3 (x-3))+C