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

The integral of 5/(x^{3/2)-25sqrt(x)} is -ln| 1/5 sqrt(x)+1|+ln| 1/5 sqrt(x)-1|+C

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integral of 5/(x^{3/2)-25sqrt(x)}

\int \frac{5}{x ^{\frac{3}{2}} - 25 x} d x

- ln \frac{1}{5} x + 1 + ln \frac{1}{5} x - 1 + C

Solution steps

\int \frac{5}{x ^{\frac{3}{2}} - 25 x} d x

Take the constant out:

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

= 5 \cdot \int \frac{1}{x ^{\frac{3}{2}} - 25 x} d x

Apply u-substitution

: \int \frac{2}{3 u ^{\frac{1}{3}} ( u - 25 u ^{\frac{1}{3}} )} d u

= 5 \cdot \int \frac{2}{3 u ^{\frac{1}{3}} ( u - 25 u ^{\frac{1}{3}} )} d u

Take the constant out:

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

= 5 \cdot \frac{2}{3} \cdot \int \frac{1}{u ^{\frac{1}{3}} ( u - 25 u ^{\frac{1}{3}} )} d u

Apply u-substitution

: \int \frac{3}{v ^{2} - 25} d v

= 5 \cdot \frac{2}{3} \cdot \int \frac{3}{v ^{2} - 25} d v

Take the constant out:

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

= 5 \cdot \frac{2}{3} \cdot 3 \cdot \int \frac{1}{v ^{2} - 25} d v

Apply Integral Substitution

= 5 \cdot \frac{2}{3} \cdot 3 \cdot \int \frac{1}{5 ( w ^{2} - 1 )} d w

Take the constant out:

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

= 5 \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{5} \cdot \int \frac{1}{w ^{2} - 1} d w

Factor

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

= 5 \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{5} \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

= 5 \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{5} (- \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}

= 5 \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{5} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

Substitute back

= 5 \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{5} - \frac{ln \frac{( x ^{\frac{3}{2}} ) ^{\frac{1}{3}}}{5} + 1}{2} - \frac{ln \frac{( x ^{\frac{3}{2}} ) ^{\frac{1}{3}}}{5} - 1}{2}

Simplify

5 \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{5} - \frac{ln \frac{( x ^{\frac{3}{2}} ) ^{\frac{1}{3}}}{5} + 1}{2} - \frac{ln \frac{( x ^{\frac{3}{2}} ) ^{\frac{1}{3}}}{5} - 1}{2} : - ln \frac{1}{5} x + 1 + ln \frac{1}{5} x - 1

= - ln \frac{1}{5} x + 1 + ln \frac{1}{5} x - 1

Add a constant to the solution

= - ln \frac{1}{5} x + 1 + ln \frac{1}{5} x - 1 + C

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What is the integral of 5/(x^{3/2)-25sqrt(x)} ?
The integral of 5/(x^{3/2)-25sqrt(x)} is -ln| 1/5 sqrt(x)+1|+ln| 1/5 sqrt(x)-1|+C