What is the integral of (27)/(x^{3/2)-81sqrt(x)} ?

The integral of (27)/(x^{3/2)-81sqrt(x)} is -6(1/2 ln|(sqrt(x))/9+1|-1/2 ln|(sqrt(x))/9-1|)+C

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integral of (27)/(x^{3/2)-81sqrt(x)}

\int \frac{27}{x ^{\frac{3}{2}} - 81 x} d x

- 6 (\frac{1}{2} ln \frac{x}{9} + 1 - \frac{1}{2} ln \frac{x}{9} - 1) + C

Solution steps

\int \frac{27}{x ^{\frac{3}{2}} - 81 x} d x

Take the constant out:

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

= 27 \cdot \int \frac{1}{x ^{\frac{3}{2}} - 81 x} d x

Apply u-substitution

: \int \frac{2}{3 u ^{\frac{1}{3}} ( u - 81 u ^{\frac{1}{3}} )} d u

= 27 \cdot \int \frac{2}{3 u ^{\frac{1}{3}} ( u - 81 u ^{\frac{1}{3}} )} d u

Take the constant out:

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

= 27 \cdot \frac{2}{3} \cdot \int \frac{1}{u ^{\frac{1}{3}} ( u - 81 u ^{\frac{1}{3}} )} d u

Apply u-substitution

: \int \frac{3}{v ^{2} - 81} d v

= 27 \cdot \frac{2}{3} \cdot \int \frac{3}{v ^{2} - 81} d v

Take the constant out:

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

= 27 \cdot \frac{2}{3} \cdot 3 \cdot \int \frac{1}{v ^{2} - 81} d v

Apply Integral Substitution

= 27 \cdot \frac{2}{3} \cdot 3 \cdot \int \frac{1}{9 ( w ^{2} - 1 )} d w

Take the constant out:

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

= 27 \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{9} \cdot \int \frac{1}{w ^{2} - 1} d w

Factor

w^{2} - 1 : - (- w^{2} + 1)

= 27 \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{9} \cdot \int \frac{1}{- ( - w ^{2} + 1 )} d w

Take the constant out:

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

= 27 \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{9} (- \int \frac{1}{- w ^{2} + 1} d w)

Use the common integral:

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

= 27 \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{9} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

Substitute back

= 27 \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{9} - \frac{ln \frac{( x ^{\frac{3}{2}} ) ^{\frac{1}{3}}}{9} + 1}{2} - \frac{ln \frac{( x ^{\frac{3}{2}} ) ^{\frac{1}{3}}}{9} - 1}{2}

Simplify

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

= - 6 (\frac{1}{2} ln \frac{x}{9} + 1 - \frac{1}{2} ln \frac{x}{9} - 1)

Add a constant to the solution

= - 6 (\frac{1}{2} ln \frac{x}{9} + 1 - \frac{1}{2} ln \frac{x}{9} - 1) + C

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

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What is the integral of (27)/(x^{3/2)-81sqrt(x)} ?
The integral of (27)/(x^{3/2)-81sqrt(x)} is -6(1/2 ln|(sqrt(x))/9+1|-1/2 ln|(sqrt(x))/9-1|)+C