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

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

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

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

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

Solution steps

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

Apply Integration By Parts

: \frac{2 - x ^{2}}{x} - \int \frac{x ^{2} - 4}{x ^{2} - x ^{2} + 2} d x

= \frac{2 - x ^{2}}{x} - \int \frac{x ^{2} - 4}{x ^{2} - x ^{2} + 2} d x

\int \frac{x ^{2} - 4}{x ^{2} - x ^{2} + 2} d x = arcsin (\frac{1}{2} x) + \frac{2 - x ^{2} + 2}{x}

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

Simplify

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

Add a constant to the solution

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

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