What is the integral of (sqrt(x^2-81))/(x^4) ?

The integral of (sqrt(x^2-81))/(x^4) is ((x^2-81)^{3/2})/(243x^3)+C

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integral of (sqrt(x^2-81))/(x^4)

integral

\frac{x ^{2} - 81}{x ^{4}}

\frac{( x ^{2} - 81 ) ^{\frac{3}{2}}}{243 x ^{3}} + C

Solution steps

\int \frac{x ^{2} - 81}{x ^{4}} d x

Apply Trigonometric Substitution

= \int \frac{tan ^{2} ( u )}{81 sec ^{3} ( 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{tan ^{2} ( u )}{sec ^{3} ( u )} d u

Rewrite using trig identities

= \frac{1}{81} \cdot \int sin^{2} (u) cos (u) d u

Apply u-substitution

= \frac{1}{81} \cdot \int v^{2} d v

Apply the Power Rule

= \frac{1}{81} \cdot \frac{v ^{3}}{3}

Substitute back

= \frac{1}{81} \cdot \frac{sin ^{3} ( arcsec ( \frac{1}{9} x ) )}{3}

Simplify

\frac{1}{81} \cdot \frac{sin ^{3} ( arcsec ( \frac{1}{9} x ) )}{3} : \frac{( x ^{2} - 81 ) ^{\frac{3}{2}}}{243 x ^{3}}

= \frac{( x ^{2} - 81 ) ^{\frac{3}{2}}}{243 x ^{3}}

Add a constant to the solution

= \frac{( x ^{2} - 81 ) ^{\frac{3}{2}}}{243 x ^{3}} + C

What is the integral of (sqrt(x^2-81))/(x^4) ?
The integral of (sqrt(x^2-81))/(x^4) is ((x^2-81)^{3/2})/(243x^3)+C