What is the integral of 1/(sqrt(7x^2-8)) ?

The integral of 1/(sqrt(7x^2-8)) is 1/(2sqrt(7))(2ln|sqrt(7x^2-8)+sqrt(7)x|-3ln(2))+C

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integral of 1/(sqrt(7x^2-8))

\int \frac{1}{7 x ^{2} - 8} d x

\frac{1}{2 7} (2 ln 7 x^{2} - 8 + 7 x - 3 ln (2)) + C

Solution steps

\int \frac{1}{7 x ^{2} - 8} d x

Apply Trigonometric Substitution

: \int \frac{sec ( u )}{7} d u

= \int \frac{sec ( u )}{7} d u

Take the constant out:

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

= \frac{1}{7} \cdot \int sec (u) d u

Use the common integral:

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

= \frac{1}{7} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = arcsec (\frac{7}{2 2} x)

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

Simplify

\frac{1}{7} ln tan (arcsec (\frac{7}{2 2} x)) + sec (arcsec (\frac{7}{2 2} x)) : \frac{1}{2 7} (2 ln 7 x^{2} - 8 + 7 x - 3 ln (2))

= \frac{1}{2 7} (2 ln 7 x^{2} - 8 + 7 x - 3 ln (2))

Add a constant to the solution

= \frac{1}{2 7} (2 ln 7 x^{2} - 8 + 7 x - 3 ln (2)) + C

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What is the integral of 1/(sqrt(7x^2-8)) ?
The integral of 1/(sqrt(7x^2-8)) is 1/(2sqrt(7))(2ln|sqrt(7x^2-8)+sqrt(7)x|-3ln(2))+C