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

The integral of (e^{x/a}+1)^{1/3}e^{x/a} is a 3/4 (e^{x/a}+1)^{4/3}+C

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integral of (e^{x/a}+1)^{1/3}e^{x/a}

\int (e^{\frac{x}{a}} + 1)^{\frac{1}{3}} e^{\frac{x}{a}} d x

a \frac{3}{4} (e^{\frac{x}{a}} + 1)^{\frac{4}{3}} + C

Solution steps

\int (e^{\frac{x}{a}} + 1)^{\frac{1}{3}} e^{\frac{x}{a}} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

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

= a \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = e^{\frac{x}{a}} + 1

= a \frac{3}{4} (e^{\frac{x}{a}} + 1)^{\frac{4}{3}}

Add a constant to the solution

= a \frac{3}{4} (e^{\frac{x}{a}} + 1)^{\frac{4}{3}} + C

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What is the integral of (e^{x/a}+1)^{1/3}e^{x/a} ?
The integral of (e^{x/a}+1)^{1/3}e^{x/a} is a 3/4 (e^{x/a}+1)^{4/3}+C