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

The integral of e^x(-e^x) is -1/2 e^{2x}+C

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integral of e^x(-e^x)

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

- \frac{1}{2} e^{2 x} + C

Solution steps

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

Simplify

e^{x} (- e^{x}) : - e^{2 x}

= \int - e^{2 x} d x

Take the constant out:

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

= - \int e^{2 x} d x

Apply u-substitution

: \int e^{u} \frac{1}{2} d u

= - \int e^{u} \frac{1}{2} d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= - \frac{1}{2} e^{u}

Substitute back

u = 2 x

= - \frac{1}{2} e^{2 x}

Add a constant to the solution

= - \frac{1}{2} e^{2 x} + C

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