What is the integral of (z^5)/((9z^2-25)^{3/2)} ?

The integral of (z^5)/((9z^2-25)^{3/2)} is (81z^4+900z^2-5000)/(2187(9z^2-25)^{1/2)}+C

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integral of (z^5)/((9z^2-25)^{3/2)}

\int \frac{z ^{5}}{( 9 z ^{2} - 25 ) ^{\frac{3}{2}}} d z

\frac{81 z ^{4} + 900 z ^{2} - 5000}{2187 ( 9 z ^{2} - 25 ) ^{\frac{1}{2}}} + C

Solution steps

\int \frac{z ^{5}}{( 9 z ^{2} - 25 ) ^{\frac{3}{2}}} d z

Apply Trigonometric Substitution

: \int \frac{125 sec ^{6} ( u )}{729 tan ^{2} ( u )} d u

= \int \frac{125 sec ^{6} ( u )}{729 tan ^{2} ( u )} d u

Take the constant out:

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

= \frac{125}{729} \cdot \int \frac{sec ^{6} ( u )}{tan ^{2} ( u )} d u

Rewrite using trig identities

= \frac{125}{729} \cdot \int \frac{1}{cos ^{4} ( u ) sin ^{2} ( u )} d u

Apply Trigonometric Substitution

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

= \frac{125}{729} \cdot \int \frac{( 1 + v ^{2} ) ^{2}}{v ^{2}} d v

Expand

\frac{( 1 + v ^{2} ) ^{2}}{v ^{2}} : \frac{1}{v ^{2}} + 2 + v^{2}

= \frac{125}{729} \cdot \int \frac{1}{v ^{2}} + 2 + 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{125}{729} (\int \frac{1}{v ^{2}} d v + \int 2 d v + \int v^{2} d v)

\int \frac{1}{v ^{2}} d v = - \frac{1}{v}

\int 2 d v = 2 v

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

= \frac{125}{729} (- \frac{1}{v} + 2 v + \frac{v ^{3}}{3})

Substitute back

= \frac{125}{729} (- \frac{1}{tan ( arcsec ( \frac{3}{5} z ) )} + 2 tan (arcsec (\frac{3}{5} z)) + \frac{tan ^{3} ( arcsec ( \frac{3}{5} z ) )}{3})

Simplify

\frac{125}{729} (- \frac{1}{tan ( arcsec ( \frac{3}{5} z ) )} + 2 tan (arcsec (\frac{3}{5} z)) + \frac{tan ^{3} ( arcsec ( \frac{3}{5} z ) )}{3}) : \frac{81 z ^{4} + 900 z ^{2} - 5000}{2187 ( 9 z ^{2} - 25 ) ^{\frac{1}{2}}}

= \frac{81 z ^{4} + 900 z ^{2} - 5000}{2187 ( 9 z ^{2} - 25 ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{81 z ^{4} + 900 z ^{2} - 5000}{2187 ( 9 z ^{2} - 25 ) ^{\frac{1}{2}}} + C

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What is the integral of (z^5)/((9z^2-25)^{3/2)} ?
The integral of (z^5)/((9z^2-25)^{3/2)} is (81z^4+900z^2-5000)/(2187(9z^2-25)^{1/2)}+C