What is the integral of (x^2-6x+9)/(sqrt(x^2-6x+45)) ?

The integral of (x^2-6x+9)/(sqrt(x^2-6x+45)) is

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integral of (x^2-6x+9)/(sqrt(x^2-6x+45))

\int \frac{x ^{2} - 6 x + 9}{x ^{2} - 6 x + 45} d x

36 (- ln (\frac{1}{6} x - 3 + x^{2} - 6 x + 45) + \frac{1}{72} (x - 3) x^{2} - 6 x + 45 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 45}{6})) + C

Solution steps

\int \frac{x ^{2} - 6 x + 9}{x ^{2} - 6 x + 45} d x

Complete the square

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

= \int \frac{( x - 3 ) ^{2}}{x ^{2} - 6 x + 45} d x

Apply u-substitution

: \int \frac{u ^{2}}{u ^{2} + 36} d u

= \int \frac{u ^{2}}{u ^{2} + 36} d u

Apply Trigonometric Substitution

: \int 36 tan^{2} (v) sec (v) d v

= \int 36 tan^{2} (v) sec (v) d v

Take the constant out:

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

= 36 \cdot \int tan^{2} (v) sec (v) d v

Rewrite using trig identities

= 36 \cdot \int (- 1 + sec^{2} (v)) sec (v) d v

Expand

(- 1 + sec^{2} (v)) sec (v) : - sec (v) + sec^{3} (v)

= 36 \cdot \int - sec (v) + sec^{3} (v) d v

Apply the Sum Rule:

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

= 36 (- \int sec (v) d v + \int sec^{3} (v) d v)

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

\int sec^{3} (v) d v = \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣

= 36 (- ln ∣ tan (v) + sec (v) ∣ + \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Substitute back

= 36 (- ln tan (arctan (\frac{1}{6} (x - 3))) + sec (arctan (\frac{1}{6} (x - 3))) + \frac{1}{2} sec (arctan (\frac{1}{6} (x - 3))) tan (arctan (\frac{1}{6} (x - 3))) + \frac{1}{2} ln tan (arctan (\frac{1}{6} (x - 3))) + sec (arctan (\frac{1}{6} (x - 3))))

Simplify

36 (- ln tan (arctan (\frac{1}{6} (x - 3))) + sec (arctan (\frac{1}{6} (x - 3))) + \frac{1}{2} sec (arctan (\frac{1}{6} (x - 3))) tan (arctan (\frac{1}{6} (x - 3))) + \frac{1}{2} ln tan (arctan (\frac{1}{6} (x - 3))) + sec (arctan (\frac{1}{6} (x - 3)))) : 36 (- ln (\frac{1}{6} x - 3 + x^{2} - 6 x + 45) + \frac{1}{72} (x - 3) x^{2} - 6 x + 45 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 45}{6}))

= 36 (- ln (\frac{1}{6} x - 3 + x^{2} - 6 x + 45) + \frac{1}{72} (x - 3) x^{2} - 6 x + 45 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 45}{6}))

Add a constant to the solution

= 36 (- ln (\frac{1}{6} x - 3 + x^{2} - 6 x + 45) + \frac{1}{72} (x - 3) x^{2} - 6 x + 45 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 45}{6})) + C

y^{^{'}} = (3 x - 2)^{2} (5 x + 4)

t^{3} - 12 t

y^{^{''}} = \frac{1}{x ^{2}} y

\int_{0}^{x} sin (t) d t

\int_{1}^{3} (x^{2} - 1) d x

What is the integral of (x^2-6x+9)/(sqrt(x^2-6x+45)) ?
The integral of (x^2-6x+9)/(sqrt(x^2-6x+45)) is