What is the integral of 1/(x^2sqrt(x^2-81)) ?

The integral of 1/(x^2sqrt(x^2-81)) is (sqrt(x^2-81))/(81x)+C

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integral of 1/(x^2sqrt(x^2-81))

\int \frac{1}{x ^{2} x ^{2} - 81} d x

\frac{x ^{2} - 81}{81 x} + C

Solution steps

\int \frac{1}{x ^{2} x ^{2} - 81} d x

Apply Trigonometric Substitution

: \int \frac{1}{81 sec ( u )} d u

= \int \frac{1}{81 sec ( u )} d u

Take the constant out:

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

= \frac{1}{81} \cdot \int \frac{1}{sec ( u )} d u

Rewrite using trig identities

= \frac{1}{81} \cdot \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= \frac{1}{81} sin (u)

Substitute back

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

= \frac{1}{81} sin (arcsec (\frac{1}{9} x))

Simplify

\frac{1}{81} sin (arcsec (\frac{1}{9} x)) : \frac{x ^{2} - 81}{81 x}

= \frac{x ^{2} - 81}{81 x}

Add a constant to the solution

= \frac{x ^{2} - 81}{81 x} + C

\int_{- 1}^{3} (- x^{3} + 3 x^{2} + 1) d x

\frac{- 8 x}{y} = 0

x \to 1 lim (x^{3} (5 - x))

x \to 1 lim (x^{\frac{1}{1 - x}})

n = 1 \sum \infty n (\frac{2}{3})^{n}

What is the integral of 1/(x^2sqrt(x^2-81)) ?
The integral of 1/(x^2sqrt(x^2-81)) is (sqrt(x^2-81))/(81x)+C