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

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

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

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

- \frac{2}{5} (4 - e^{x})^{\frac{5}{2}} + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

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

= - \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = 4 - e^{x}

= - \frac{2}{5} (4 - e^{x})^{\frac{5}{2}}

Add a constant to the solution

= - \frac{2}{5} (4 - e^{x})^{\frac{5}{2}} + C

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