What is the integral of (e^{2t}+2)^{1/3}e^{2t} ?

The integral of (e^{2t}+2)^{1/3}e^{2t} is 3/8 (e^{2t}+2)^{4/3}+C

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integral of (e^{2t}+2)^{1/3}e^{2t}

\int (e^{2 t} + 2)^{\frac{1}{3}} e^{2 t} d t

\frac{3}{8} (e^{2 t} + 2)^{\frac{4}{3}} + C

Solution steps

\int (e^{2 t} + 2)^{\frac{1}{3}} e^{2 t} d t

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: \frac{3}{4} u^{\frac{4}{3}}

= \frac{1}{2} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = e^{2 t} + 2

= \frac{1}{2} \cdot \frac{3}{4} (e^{2 t} + 2)^{\frac{4}{3}}

Simplify

\frac{1}{2} \cdot \frac{3}{4} (e^{2 t} + 2)^{\frac{4}{3}} : \frac{3}{8} (e^{2 t} + 2)^{\frac{4}{3}}

= \frac{3}{8} (e^{2 t} + 2)^{\frac{4}{3}}

Add a constant to the solution

= \frac{3}{8} (e^{2 t} + 2)^{\frac{4}{3}} + C

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What is the integral of (e^{2t}+2)^{1/3}e^{2t} ?
The integral of (e^{2t}+2)^{1/3}e^{2t} is 3/8 (e^{2t}+2)^{4/3}+C