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integral of ((9-4x^2)^{1/2})/(x^6)

Solution

\int \frac{( 9 - 4 x ^{2} ) ^{\frac{1}{2}}}{x ^{6}} d x

- \frac{32}{81} \frac{( 9 - 4 x ^{2} ) ^{\frac{5}{2}}}{160 x ^{5}} + \frac{( 9 - 4 x ^{2} ) ^{\frac{3}{2}}}{24 x ^{3}} + C

Solution steps

\int \frac{( 9 - 4 x ^{2} ) ^{\frac{1}{2}}}{x ^{6}} d x

Apply Trigonometric Substitution

: \int \frac{32 cos ^{2} ( u )}{81 sin ^{6} ( u )} d u

= \int \frac{32 cos ^{2} ( u )}{81 sin ^{6} ( u )} d u

Take the constant out:

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

= \frac{32}{81} \cdot \int \frac{cos ^{2} ( u )}{sin ^{6} ( u )} d u

Rewrite using trig identities

= \frac{32}{81} \cdot \int cot^{2} (u) csc^{4} (u) d u

Apply Trigonometric Substitution

: \int - v^{2} (v^{2} + 1) d v

= \frac{32}{81} \cdot \int - v^{2} (v^{2} + 1) d v

Take the constant out:

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

= \frac{32}{81} (- \int v^{2} (v^{2} + 1) d v)

Expand

v^{2} (v^{2} + 1) : v^{4} + v^{2}

= \frac{32}{81} (- \int v^{4} + v^{2} d v)

Apply the Sum Rule:

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

= \frac{32}{81} (- (\int v^{4} d v + \int v^{2} d v))

\int v^{4} d v = \frac{v ^{5}}{5}

\int v^{2} d v = \frac{v ^{3}}{3}

= \frac{32}{81} (- (\frac{v ^{5}}{5} + \frac{v ^{3}}{3}))

Substitute back

= \frac{32}{81} (- (\frac{cot ^{5} ( arcsin ( \frac{2}{3} x ) )}{5} + \frac{cot ^{3} ( arcsin ( \frac{2}{3} x ) )}{3}))

Simplify

\frac{32}{81} (- (\frac{cot ^{5} ( arcsin ( \frac{2}{3} x ) )}{5} + \frac{cot ^{3} ( arcsin ( \frac{2}{3} x ) )}{3})) : - \frac{32}{81} \frac{( 9 - 4 x ^{2} ) ^{\frac{5}{2}}}{160 x ^{5}} + \frac{( 9 - 4 x ^{2} ) ^{\frac{3}{2}}}{24 x ^{3}}

= - \frac{32}{81} \frac{( 9 - 4 x ^{2} ) ^{\frac{5}{2}}}{160 x ^{5}} + \frac{( 9 - 4 x ^{2} ) ^{\frac{3}{2}}}{24 x ^{3}}

Add a constant to the solution

= - \frac{32}{81} \frac{( 9 - 4 x ^{2} ) ^{\frac{5}{2}}}{160 x ^{5}} + \frac{( 9 - 4 x ^{2} ) ^{\frac{3}{2}}}{24 x ^{3}} + C

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