What is the integral of 8/(9xsqrt(81x^2-64)) ?

The integral of 8/(9xsqrt(81x^2-64)) is 1/9 arcsec(9/8 x)+C

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integral of 8/(9xsqrt(81x^2-64))

\int \frac{8}{9 x 81 x ^{2} - 64} d x

\frac{1}{9} arcsec (\frac{9}{8} x) + C

Solution steps

\int \frac{8}{9 x 81 x ^{2} - 64} d x

Take the constant out:

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

= \frac{8}{9} \cdot \int \frac{1}{x 81 x ^{2} - 64} d x

Apply Trigonometric Substitution

= \frac{8}{9} \cdot \int \frac{1}{8} d u

Take the constant out:

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

= \frac{8}{9} \cdot \frac{1}{8} \cdot \int 1 d u

Integral of a constant:

\int a d x = a x

= \frac{8}{9} \cdot \frac{1}{8} \cdot 1 \cdot u

Substitute back

u = arcsec (\frac{9}{8} x)

= \frac{8}{9} \cdot \frac{1}{8} \cdot 1 \cdot arcsec (\frac{9}{8} x)

Simplify

\frac{8}{9} \cdot \frac{1}{8} \cdot 1 \cdot arcsec (\frac{9}{8} x) : \frac{1}{9} arcsec (\frac{9}{8} x)

= \frac{1}{9} arcsec (\frac{9}{8} x)

Add a constant to the solution

= \frac{1}{9} arcsec (\frac{9}{8} x) + C

What is the integral of 8/(9xsqrt(81x^2-64)) ?
The integral of 8/(9xsqrt(81x^2-64)) is 1/9 arcsec(9/8 x)+C