What is the integral of (sqrt(x-7))/(x+6) ?

The integral of (sqrt(x-7))/(x+6) is 2sqrt(13)(-arctan(sqrt(1/13 (x-7)))+sqrt(1/13 (x-7)))+C

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integral of (sqrt(x-7))/(x+6)

\int \frac{x - 7}{x + 6} d x

213 (- arctan (\frac{1}{13} (x - 7)) + \frac{1}{13} (x - 7)) + C

Solution steps

\int \frac{x - 7}{x + 6} d x

Apply u-substitution

: \int \frac{2 u ^{2}}{u ^{2} + 13} d u

= \int \frac{2 u ^{2}}{u ^{2} + 13} d u

Take the constant out:

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

= 2 \cdot \int \frac{u ^{2}}{u ^{2} + 13} d u

\frac{u ^{2}}{u ^{2} + 13} = - \frac{13}{u ^{2} + 13} + 1

= 2 \cdot \int - \frac{13}{u ^{2} + 13} + 1 d u

Apply Integral Substitution

= 2 \cdot \int 13 (- \frac{1}{v ^{2} + 1} + 1) d v

Take the constant out:

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

= 213 \cdot \int - \frac{1}{v ^{2} + 1} + 1 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

= 213 (- \int \frac{1}{v ^{2} + 1} d v + \int 1 d v)

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

\int 1 d v = v

= 213 (- arctan (v) + v)

Substitute back

= 213 (- arctan (\frac{x - 7}{13}) + \frac{x - 7}{13})

Simplify

213 (- arctan (\frac{x - 7}{13}) + \frac{x - 7}{13}) : 213 (- arctan (\frac{1}{13} (x - 7)) + \frac{1}{13} (x - 7))

= 213 (- arctan (\frac{1}{13} (x - 7)) + \frac{1}{13} (x - 7))

Add a constant to the solution

= 213 (- arctan (\frac{1}{13} (x - 7)) + \frac{1}{13} (x - 7)) + C

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What is the integral of (sqrt(x-7))/(x+6) ?
The integral of (sqrt(x-7))/(x+6) is 2sqrt(13)(-arctan(sqrt(1/13 (x-7)))+sqrt(1/13 (x-7)))+C