What is the integral of 9/(x^2sqrt(81-x^2)) ?

The integral of 9/(x^2sqrt(81-x^2)) is -(sqrt(81-x^2))/(9x)+C

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integral of 9/(x^2sqrt(81-x^2))

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

- \frac{81 - x ^{2}}{9 x} + C

Solution steps

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

Take the constant out:

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

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

Apply Trigonometric Substitution

= 9 \cdot \int \frac{cot ( u )}{81 cos ( u ) sin ( u )} d u

Take the constant out:

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

= 9 \cdot \frac{1}{81} \cdot \int \frac{cot ( u )}{cos ( u ) sin ( u )} d u

Simplify

\frac{cot ( u )}{cos ( u ) sin ( u )} : csc^{2} (u)

= 9 \cdot \frac{1}{81} \cdot \int csc^{2} (u) d u

Use the common integral:

\int csc^{2} (u) d u = - cot (u)

= 9 \cdot \frac{1}{81} (- cot (u))

Substitute back

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

= 9 \cdot \frac{1}{81} (- cot (arcsin (\frac{1}{9} x)))

Simplify

9 \cdot \frac{1}{81} (- cot (arcsin (\frac{1}{9} x))) : - \frac{81 - x ^{2}}{9 x}

= - \frac{81 - x ^{2}}{9 x}

Add a constant to the solution

= - \frac{81 - x ^{2}}{9 x} + C

What is the integral of 9/(x^2sqrt(81-x^2)) ?
The integral of 9/(x^2sqrt(81-x^2)) is -(sqrt(81-x^2))/(9x)+C