What is the integral of 1/((5x-2)^{3/2)} ?

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

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integral of 1/((5x-2)^{3/2)}

\int \frac{1}{( 5 x - 2 ) ^{\frac{3}{2}}} d x

- \frac{2}{5 ( 5 x - 2 ) ^{\frac{1}{2}}} + C

Solution steps

\int \frac{1}{( 5 x - 2 ) ^{\frac{3}{2}}} d x

Apply u-substitution

: \int \frac{1}{5 u ^{\frac{3}{2}}} d u

= \int \frac{1}{5 u ^{\frac{3}{2}}} d u

Take the constant out:

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

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

Apply the Power Rule

: - \frac{2}{u ^{\frac{1}{2}}}

= \frac{1}{5} (- \frac{2}{u ^{\frac{1}{2}}})

Substitute back

u = 5 x - 2

= \frac{1}{5} (- \frac{2}{( 5 x - 2 ) ^{\frac{1}{2}}})

Simplify

\frac{1}{5} (- \frac{2}{( 5 x - 2 ) ^{\frac{1}{2}}}) : - \frac{2}{5 ( 5 x - 2 ) ^{\frac{1}{2}}}

= - \frac{2}{5 ( 5 x - 2 ) ^{\frac{1}{2}}}

Add a constant to the solution

= - \frac{2}{5 ( 5 x - 2 ) ^{\frac{1}{2}}} + C

x \to 2 lim (sin (x - 2) + 1)

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\int_{0}^{1} \frac{80}{x ^{2} - 100} d x

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