What is the integral of ex(1+e^x)(1/2) ?

The integral of ex(1+e^x)(1/2) is e/4 (x^2+2e^xx-2e^x)+C

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integral of ex(1+e^x)(1/2)

\int e x (1 + e^{x}) (\frac{1}{2}) d x

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

Solution steps

\int e x (1 + e^{x}) \frac{1}{2} d x

Take the constant out:

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

= e \frac{1}{2} \cdot \int x (1 + e^{x}) d x

e \frac{1}{2} = \frac{e}{2}

= \frac{e}{2} \cdot \int x (1 + e^{x}) d x

Apply Integration By Parts

: x (x + e^{x}) - \int x + e^{x} d x

= \frac{e}{2} (x (x + e^{x}) - \int x + e^{x} d x)

\int x + e^{x} d x = \frac{x ^{2}}{2} + e^{x}

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of ex(1+e^x)(1/2) ?
The integral of ex(1+e^x)(1/2) is e/4 (x^2+2e^xx-2e^x)+C