What is the integral of 2sqrt((4-x^2)/(x^8)) ?

The integral of 2sqrt((4-x^2)/(x^8)) is 2(-(sqrt(4-x^2))/(3x^3)+(sqrt(4-x^2))/(12x))+C

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integral of 2sqrt((4-x^2)/(x^8))

\int 2 \frac{4 - x ^{2}}{x ^{8}} d x

2 (- \frac{4 - x ^{2}}{3 x ^{3}} + \frac{4 - x ^{2}}{12 x}) + C

Solution steps

\int 2 \frac{4 - x ^{2}}{x ^{8}} d x

Take the constant out:

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

= 2 \cdot \int \frac{4 - x ^{2}}{x ^{8}} d x

Simplify

\frac{4 - x ^{2}}{x ^{8}} : \frac{4 - x ^{2}}{x ^{4}}

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

Apply Integration By Parts

: - \frac{4 - x ^{2}}{3 x ^{3}} - \int \frac{1}{3 x ^{2} - x ^{2} + 4} d x

= 2 (- \frac{4 - x ^{2}}{3 x ^{3}} - \int \frac{1}{3 x ^{2} - x ^{2} + 4} d x)

\int \frac{1}{3 x ^{2} - x ^{2} + 4} d x = - \frac{4 - x ^{2}}{12 x}

= 2 (- \frac{4 - x ^{2}}{3 x ^{3}} - (- \frac{4 - x ^{2}}{12 x}))

Simplify

= 2 (- \frac{4 - x ^{2}}{3 x ^{3}} + \frac{4 - x ^{2}}{12 x})

Add a constant to the solution

= 2 (- \frac{4 - x ^{2}}{3 x ^{3}} + \frac{4 - x ^{2}}{12 x}) + C

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What is the integral of 2sqrt((4-x^2)/(x^8)) ?
The integral of 2sqrt((4-x^2)/(x^8)) is 2(-(sqrt(4-x^2))/(3x^3)+(sqrt(4-x^2))/(12x))+C