What is the integral of 1/(sqrt(x)(\sqrt{x-7))} ?

The integral of 1/(sqrt(x)(\sqrt{x-7))} is 2ln| 1/(sqrt(7))sqrt(x-7)+sqrt(x/7)|+C

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integral of 1/(sqrt(x)(\sqrt{x-7))}

\int \frac{1}{x ( x - 7 )} d x

2 ln \frac{1}{7} x - 7 + \frac{x}{7} + C

Solution steps

\int \frac{1}{x x - 7} d x

Apply u-substitution

: \int \frac{2}{u ^{2} + 7} d u

= \int \frac{2}{u ^{2} + 7} d u

Take the constant out:

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

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

Apply Trigonometric Substitution

: \int sec (v) d v

= 2 \cdot \int sec (v) d v

Use the common integral:

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

= 2 ln ∣ tan (v) + sec (v) ∣

Substitute back

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

Simplify

2 ln tan (arctan (\frac{1}{7} x - 7)) + sec (arctan (\frac{1}{7} x - 7)) : 2 ln \frac{1}{7} x - 7 + \frac{x}{7}

= 2 ln \frac{1}{7} x - 7 + \frac{x}{7}

Add a constant to the solution

= 2 ln \frac{1}{7} x - 7 + \frac{x}{7} + C

- tan (x)

\frac{d}{d x} (cos (x) + 2 x^{2} + 1)

\frac{1}{( s ^{2} - 6 s + 10 )}

\frac{\partial ^{2}}{\partial x ^{2}} (e^{3 x + 4 y} cos (5 z))

\frac{2}{5 ( s + 3 ) ^{2} + 125}

What is the integral of 1/(sqrt(x)(\sqrt{x-7))} ?
The integral of 1/(sqrt(x)(\sqrt{x-7))} is 2ln| 1/(sqrt(7))sqrt(x-7)+sqrt(x/7)|+C