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

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

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

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

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

Solution steps

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

Take the constant out:

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

= 4 \cdot \int e^{x} 4 + e^{x} d x

Apply u-substitution

= 4 \cdot \int u d u

Apply the Power Rule

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

Substitute back

u = 4 + e^{x}

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

Simplify

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

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

Add a constant to the solution

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

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