What is the integral of sqrt((x^3-6)/(x^{11))} ?

The integral of sqrt((x^3-6)/(x^{11))} is ((x^3-6)^{3/2})/(27x^{9/2)}+C

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integral of sqrt((x^3-6)/(x^{11))}

\int \frac{x ^{3} - 6}{x ^{11}} d x

\frac{( x ^{3} - 6 ) ^{\frac{3}{2}}}{27 x ^{\frac{9}{2}}} + C

Solution steps

\int \frac{x ^{3} - 6}{x ^{11}} d x

Simplify

\frac{x ^{3} - 6}{x ^{11}} : \frac{x ^{3} - 6}{x ^{11}}

= \int \frac{x ^{3} - 6}{x ^{11}} d x

Apply u-substitution

: \int \frac{u}{3 ( u + 6 ) ^{\frac{5}{2}}} d u

= \int \frac{u}{3 ( u + 6 ) ^{\frac{5}{2}}} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{u}{( u + 6 ) ^{\frac{5}{2}}} d u

Apply u-substitution

: \int \frac{2 v ^{2}}{( v ^{2} + 6 ) ^{\frac{5}{2}}} d v

= \frac{1}{3} \cdot \int \frac{2 v ^{2}}{( v ^{2} + 6 ) ^{\frac{5}{2}}} d v

Take the constant out:

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

= \frac{1}{3} \cdot 2 \cdot \int \frac{v ^{2}}{( v ^{2} + 6 ) ^{\frac{5}{2}}} d v

Apply Trigonometric Substitution

: \int \frac{tan ^{2} ( w )}{6 sec ^{3} ( w )} d w

= \frac{1}{3} \cdot 2 \cdot \int \frac{tan ^{2} ( w )}{6 sec ^{3} ( w )} d w

Take the constant out:

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

= \frac{1}{3} \cdot 2 \cdot \frac{1}{6} \cdot \int \frac{tan ^{2} ( w )}{sec ^{3} ( w )} d w

Rewrite using trig identities

= \frac{1}{3} \cdot 2 \cdot \frac{1}{6} \cdot \int sin^{2} (w) cos (w) d w

Apply u-substitution

: \int t^{2} d t

= \frac{1}{3} \cdot 2 \cdot \frac{1}{6} \cdot \int t^{2} d t

Apply the Power Rule

: \frac{t ^{3}}{3}

= \frac{1}{3} \cdot 2 \cdot \frac{1}{6} \cdot \frac{t ^{3}}{3}

Substitute back

= \frac{1}{3} \cdot 2 \cdot \frac{1}{6} \cdot \frac{sin ^{3} ( arctan ( \frac{1}{6} x ^{3} - 6 ) )}{3}

Simplify

\frac{1}{3} \cdot 2 \cdot \frac{1}{6} \cdot \frac{sin ^{3} ( arctan ( \frac{1}{6} x ^{3} - 6 ) )}{3} : \frac{( x ^{3} - 6 ) ^{\frac{3}{2}}}{27 x ^{\frac{9}{2}}}

= \frac{( x ^{3} - 6 ) ^{\frac{3}{2}}}{27 x ^{\frac{9}{2}}}

Add a constant to the solution

= \frac{( x ^{3} - 6 ) ^{\frac{3}{2}}}{27 x ^{\frac{9}{2}}} + C

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What is the integral of sqrt((x^3-6)/(x^{11))} ?
The integral of sqrt((x^3-6)/(x^{11))} is ((x^3-6)^{3/2})/(27x^{9/2)}+C