What is the integral of (5x-4)/(sqrt(x^2+12x-64)) ?

The integral of (5x-4)/(sqrt(x^2+12x-64)) is 5sqrt(x^2+12x-64)-34ln(1/10 |sqrt(x^2+12x-64)+x+6|)+C

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integral of (5x-4)/(sqrt(x^2+12x-64))

\int \frac{5 x - 4}{x ^{2} + 12 x - 64} d x

5 x^{2} + 12 x - 64 - 34 ln (\frac{1}{10} x^{2} + 12 x - 64 + x + 6) + C

Solution steps

\int \frac{5 x - 4}{x ^{2} + 12 x - 64} d x

Expand

\frac{5 x - 4}{x ^{2} + 12 x - 64} : \frac{5 x}{x ^{2} + 12 x - 64} - \frac{4}{x ^{2} + 12 x - 64}

= \int \frac{5 x}{x ^{2} + 12 x - 64} - \frac{4}{x ^{2} + 12 x - 64} d x

Apply the Sum Rule:

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

= \int \frac{5 x}{x ^{2} + 12 x - 64} d x - \int \frac{4}{x ^{2} + 12 x - 64} d x

\int \frac{5 x}{x ^{2} + 12 x - 64} d x = 5 (x^{2} + 12 x - 64 - 6 ln (\frac{1}{10} x^{2} + 12 x - 64 + x + 6))

\int \frac{4}{x ^{2} + 12 x - 64} d x = 4 ln (\frac{1}{10} x^{2} + 12 x - 64 + x + 6)

= 5 (x^{2} + 12 x - 64 - 6 ln (\frac{1}{10} x^{2} + 12 x - 64 + x + 6)) - 4 ln (\frac{1}{10} x^{2} + 12 x - 64 + x + 6)

Simplify

5 (x^{2} + 12 x - 64 - 6 ln (\frac{1}{10} x^{2} + 12 x - 64 + x + 6)) - 4 ln (\frac{1}{10} x^{2} + 12 x - 64 + x + 6) : 5 x^{2} + 12 x - 64 - 34 ln (\frac{1}{10} x^{2} + 12 x - 64 + x + 6)

= 5 x^{2} + 12 x - 64 - 34 ln (\frac{1}{10} x^{2} + 12 x - 64 + x + 6)

Add a constant to the solution

= 5 x^{2} + 12 x - 64 - 34 ln (\frac{1}{10} x^{2} + 12 x - 64 + x + 6) + C

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What is the integral of (5x-4)/(sqrt(x^2+12x-64)) ?
The integral of (5x-4)/(sqrt(x^2+12x-64)) is 5sqrt(x^2+12x-64)-34ln(1/10 |sqrt(x^2+12x-64)+x+6|)+C