What is the integral of e^xsqrt(4-e^x) ?

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

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integral of e^xsqrt(4-e^x)

\int e^{x} 4 - e^{x} d x

- \frac{2}{3} (4 - e^{x})^{\frac{3}{2}} + C

Solution steps

\int e^{x} 4 - e^{x} d x

Apply u-substitution

: \int - u d u

= \int - u d u

Take the constant out:

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

= - \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

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

Substitute back

u = 4 - e^{x}

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

Add a constant to the solution

= - \frac{2}{3} (4 - e^{x})^{\frac{3}{2}} + C

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\frac{d}{d x} (\frac{8 x}{7 - cot ( x )})

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y^{^{''}} + 9 y = t e^{2 t}

\frac{d}{d x} (x^{9} ln (4 x))

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