What is the integral of sqrt(25-8x+x^2) ?

The integral of sqrt(25-8x+x^2) is 9(1/18 (x-4)sqrt(x^2-8x+25)+1/2 ln((|x-4+sqrt(x^2-8x+25)|)/3))+C

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integral of sqrt(25-8x+x^2)

\int 25 - 8 x + x^{2} d x

9 (\frac{1}{18} (x - 4) x^{2} - 8 x + 25 + \frac{1}{2} ln (\frac{x - 4 + x ^{2} - 8 x + 25}{3})) + C

Solution steps

\int 25 - 8 x + x^{2} d x

Complete the square

25 - 8 x + x^{2} : (x - 4)^{2} + 9

= \int (x - 4)^{2} + 9 d x

Apply u-substitution

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

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

Apply Trigonometric Substitution

: \int 9 sec^{3} (v) d v

= \int 9 sec^{3} (v) d v

Take the constant out:

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

= 9 \cdot \int sec^{3} (v) d v

Apply Integral Reduction

= 9 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} \cdot \int sec (v) d v)

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

= 9 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Substitute back

= 9 (\frac{sec ^{2} ( arctan ( \frac{1}{3} ( x - 4 ) ) ) sin ( arctan ( \frac{1}{3} ( x - 4 ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{3} (x - 4))) + sec (arctan (\frac{1}{3} (x - 4))))

Simplify

9 (\frac{sec ^{2} ( arctan ( \frac{1}{3} ( x - 4 ) ) ) sin ( arctan ( \frac{1}{3} ( x - 4 ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{3} (x - 4))) + sec (arctan (\frac{1}{3} (x - 4)))) : 9 (\frac{1}{18} (x - 4) x^{2} - 8 x + 25 + \frac{1}{2} ln (\frac{x - 4 + x ^{2} - 8 x + 25}{3}))

= 9 (\frac{1}{18} (x - 4) x^{2} - 8 x + 25 + \frac{1}{2} ln (\frac{x - 4 + x ^{2} - 8 x + 25}{3}))

Add a constant to the solution

= 9 (\frac{1}{18} (x - 4) x^{2} - 8 x + 25 + \frac{1}{2} ln (\frac{x - 4 + x ^{2} - 8 x + 25}{3})) + C

\frac{d}{d x} (2 x^{3} - 4 x - 8 x + 6)

\frac{1}{s ^{2}} (1 - e^{- 5 s})

12 x^{3} + 24 x^{2}

\frac{\partial}{\partial y} (x^{2} - 4 x y + 4 y^{2} + 2)

n = 1 \sum \infty cos (\frac{2}{n})

What is the integral of sqrt(25-8x+x^2) ?
The integral of sqrt(25-8x+x^2) is 9(1/18 (x-4)sqrt(x^2-8x+25)+1/2 ln((|x-4+sqrt(x^2-8x+25)|)/3))+C