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

The integral of 2/3 (x+3)^{3/2} is 4/15 (x+3)^{5/2}+C

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

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

\frac{4}{15} (x + 3)^{\frac{5}{2}} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

= \frac{2}{3} \cdot \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = x + 3

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

Simplify

\frac{2}{3} \cdot \frac{2}{5} (x + 3)^{\frac{5}{2}} : \frac{4}{15} (x + 3)^{\frac{5}{2}}

= \frac{4}{15} (x + 3)^{\frac{5}{2}}

Add a constant to the solution

= \frac{4}{15} (x + 3)^{\frac{5}{2}} + C

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What is the integral of 2/3 (x+3)^{3/2} ?
The integral of 2/3 (x+3)^{3/2} is 4/15 (x+3)^{5/2}+C