What is the integral of (14x-13)/(sqrt(x^2-4x-21)) ?

The integral of (14x-13)/(sqrt(x^2-4x-21)) is 14sqrt(x^2-4x-21)+15ln(1/5 |sqrt(x^2-4x-21)+x-2|)+C

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integral of (14x-13)/(sqrt(x^2-4x-21))

\int \frac{14 x - 13}{x ^{2} - 4 x - 21} d x

14 x^{2} - 4 x - 21 + 15 ln (\frac{1}{5} x^{2} - 4 x - 21 + x - 2) + C

Solution steps

\int \frac{14 x - 13}{x ^{2} - 4 x - 21} d x

Expand

\frac{14 x - 13}{x ^{2} - 4 x - 21} : \frac{14 x}{x ^{2} - 4 x - 21} - \frac{13}{x ^{2} - 4 x - 21}

= \int \frac{14 x}{x ^{2} - 4 x - 21} - \frac{13}{x ^{2} - 4 x - 21} 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{14 x}{x ^{2} - 4 x - 21} d x - \int \frac{13}{x ^{2} - 4 x - 21} d x

\int \frac{14 x}{x ^{2} - 4 x - 21} d x = 14 (x^{2} - 4 x - 21 + 2 ln (\frac{1}{5} x^{2} - 4 x - 21 + x - 2))

\int \frac{13}{x ^{2} - 4 x - 21} d x = 13 ln (\frac{1}{5} x^{2} - 4 x - 21 + x - 2)

= 14 (x^{2} - 4 x - 21 + 2 ln (\frac{1}{5} x^{2} - 4 x - 21 + x - 2)) - 13 ln (\frac{1}{5} x^{2} - 4 x - 21 + x - 2)

Simplify

14 (x^{2} - 4 x - 21 + 2 ln (\frac{1}{5} x^{2} - 4 x - 21 + x - 2)) - 13 ln (\frac{1}{5} x^{2} - 4 x - 21 + x - 2) : 14 x^{2} - 4 x - 21 + 15 ln (\frac{1}{5} x^{2} - 4 x - 21 + x - 2)

= 14 x^{2} - 4 x - 21 + 15 ln (\frac{1}{5} x^{2} - 4 x - 21 + x - 2)

Add a constant to the solution

= 14 x^{2} - 4 x - 21 + 15 ln (\frac{1}{5} x^{2} - 4 x - 21 + x - 2) + C

\frac{\partial}{\partial x} (\frac{7 y}{x ^{2} + y ^{2} + 19})

- \frac{18}{( x - 9 ) ^{2}}

f (x) = 22 x

y = - 6 (4 x^{2} + 4)^{- 6}

\int_{0}^{\infty} arctan (x) d x

What is the integral of (14x-13)/(sqrt(x^2-4x-21)) ?
The integral of (14x-13)/(sqrt(x^2-4x-21)) is 14sqrt(x^2-4x-21)+15ln(1/5 |sqrt(x^2-4x-21)+x-2|)+C