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

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

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integral of (sqrt(x))/((2+x^{3/2))^3}

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

- \frac{1}{3 ( 2 + x ^{\frac{3}{2}} ) ^{2}} + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{3 v ^{3}} d v

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

Take the constant out:

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

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

Apply the Power Rule

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

= 2 \cdot \frac{1}{3} (- \frac{1}{2 v ^{2}})

Substitute back

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

Simplify

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

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

Add a constant to the solution

= - \frac{1}{3 ( 2 + x ^{\frac{3}{2}} ) ^{2}} + C

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