What is the integral of e^x(4-e^x)^{2/3} ?

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

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integral of e^x(4-e^x)^{2/3}

\int e^{x} (4 - e^{x})^{\frac{2}{3}} d x

- \frac{3}{5} (4 - e^{x})^{\frac{5}{3}} + C

Solution steps

\int e^{x} (4 - e^{x})^{\frac{2}{3}} d x

Apply u-substitution

: \int (4 - u)^{\frac{2}{3}} d u

= \int (4 - u)^{\frac{2}{3}} d u

Apply u-substitution

: \int - v^{\frac{2}{3}} d v

= \int - v^{\frac{2}{3}} d v

Take the constant out:

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

= - \int v^{\frac{2}{3}} d v

Apply the Power Rule

: \frac{3}{5} v^{\frac{5}{3}}

= - \frac{3}{5} v^{\frac{5}{3}}

Substitute back

= - \frac{3}{5} (4 - e^{x})^{\frac{5}{3}}

Add a constant to the solution

= - \frac{3}{5} (4 - e^{x})^{\frac{5}{3}} + C

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