What is the integral of 7/(sqrt(x^2-12x+34)) ?

The integral of 7/(sqrt(x^2-12x+34)) is 7ln|sqrt(1/2 (x^2-12x+34))+1/(sqrt(2))(x-6)|+C

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integral of 7/(sqrt(x^2-12x+34))

\int \frac{7}{x ^{2} - 12 x + 34} d x

7 ln \frac{1}{2} (x^{2} - 12 x + 34) + \frac{1}{2} (x - 6) + C

Solution steps

\int \frac{7}{x ^{2} - 12 x + 34} d x

Take the constant out:

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

= 7 \cdot \int \frac{1}{x ^{2} - 12 x + 34} d x

Complete the square

x^{2} - 12 x + 34 : (x - 6)^{2} - 2

= 7 \cdot \int \frac{1}{( x - 6 ) ^{2} - 2} d x

Apply u-substitution

: \int \frac{1}{u ^{2} - 2} d u

= 7 \cdot \int \frac{1}{u ^{2} - 2} d u

Apply Trigonometric Substitution

: \int sec (v) d v

= 7 \cdot \int sec (v) d v

Use the common integral:

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

= 7 ln ∣ tan (v) + sec (v) ∣

Substitute back

= 7 ln tan (arcsec (\frac{1}{2} (x - 6))) + sec (arcsec (\frac{1}{2} (x - 6)))

Simplify

7 ln tan (arcsec (\frac{1}{2} (x - 6))) + sec (arcsec (\frac{1}{2} (x - 6))) : 7 ln \frac{1}{2} (x^{2} - 12 x + 34) + \frac{1}{2} (x - 6)

= 7 ln \frac{1}{2} (x^{2} - 12 x + 34) + \frac{1}{2} (x - 6)

Add a constant to the solution

= 7 ln \frac{1}{2} (x^{2} - 12 x + 34) + \frac{1}{2} (x - 6) + C

(- 4, 3), (- 2, 1)

x \to 7 - lim (\frac{9 x}{x - 7})

\frac{\partial}{\partial x} (tan^{2} (x))

e^{- x} sin (x)

x \to \infty lim (\frac{4 x + 1}{12 x ^{2} - 7})

What is the integral of 7/(sqrt(x^2-12x+34)) ?
The integral of 7/(sqrt(x^2-12x+34)) is 7ln|sqrt(1/2 (x^2-12x+34))+1/(sqrt(2))(x-6)|+C