What is the integral of (3x+8)^{2/3} ?

The integral of (3x+8)^{2/3} is 1/5 (3x+8)^{5/3}+C

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integral of (3x+8)^{2/3}

\int (3 x + 8)^{\frac{2}{3}} d x

\frac{1}{5} (3 x + 8)^{\frac{5}{3}} + C

Solution steps

\int (3 x + 8)^{\frac{2}{3}} d x

Apply u-substitution

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

Take the constant out:

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

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

Apply the Power Rule

= \frac{1}{3} \cdot \frac{3}{5} u^{\frac{5}{3}}

Substitute back

u = 3 x + 8

= \frac{1}{3} \cdot \frac{3}{5} (3 x + 8)^{\frac{5}{3}}

Simplify

\frac{1}{3} \cdot \frac{3}{5} (3 x + 8)^{\frac{5}{3}} : \frac{1}{5} (3 x + 8)^{\frac{5}{3}}

= \frac{1}{5} (3 x + 8)^{\frac{5}{3}}

Add a constant to the solution

= \frac{1}{5} (3 x + 8)^{\frac{5}{3}} + C

What is the integral of (3x+8)^{2/3} ?
The integral of (3x+8)^{2/3} is 1/5 (3x+8)^{5/3}+C