What is the integral of 2/((6-x)^{3/2)} ?

The integral of 2/((6-x)^{3/2)} is 4/((6-x)^{1/2)}+C

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integral of 2/((6-x)^{3/2)}

\int \frac{2}{( 6 - x ) ^{\frac{3}{2}}} d x

\frac{4}{( 6 - x ) ^{\frac{1}{2}}} + C

Solution steps

\int \frac{2}{( 6 - x ) ^{\frac{3}{2}}} d x

Take the constant out:

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

= 2 \cdot \int \frac{1}{( 6 - x ) ^{\frac{3}{2}}} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = 6 - x

= 2 (- (- \frac{2}{( 6 - x ) ^{\frac{1}{2}}}))

Simplify

2 (- (- \frac{2}{( 6 - x ) ^{\frac{1}{2}}})) : \frac{4}{( 6 - x ) ^{\frac{1}{2}}}

= \frac{4}{( 6 - x ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{4}{( 6 - x ) ^{\frac{1}{2}}} + C

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What is the integral of 2/((6-x)^{3/2)} ?
The integral of 2/((6-x)^{3/2)} is 4/((6-x)^{1/2)}+C