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

The integral of ((sqrt(9-x^2))/(x^2)) is -arcsin(1/3 x)-(sqrt(9-x^2))/x+C

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

\int (\frac{9 - x ^{2}}{x ^{2}}) d x

- arcsin (\frac{1}{3} x) - \frac{9 - x ^{2}}{x} + C

Solution steps

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

Apply Trigonometric Substitution

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

Rewrite using trig identities

= \int - 1 + csc^{2} (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int 1 d u + \int csc^{2} (u) d u

\int 1 d u = u

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

= - u - cot (u)

Substitute back

u = arcsin (\frac{1}{3} x)

= - arcsin (\frac{1}{3} x) - cot (arcsin (\frac{1}{3} x))

Simplify

- arcsin (\frac{1}{3} x) - cot (arcsin (\frac{1}{3} x)) : - arcsin (\frac{1}{3} x) - \frac{9 - x ^{2}}{x}

= - arcsin (\frac{1}{3} x) - \frac{9 - x ^{2}}{x}

Add a constant to the solution

= - arcsin (\frac{1}{3} x) - \frac{9 - x ^{2}}{x} + C

What is the integral of ((sqrt(9-x^2))/(x^2)) ?
The integral of ((sqrt(9-x^2))/(x^2)) is -arcsin(1/3 x)-(sqrt(9-x^2))/x+C