What is the integral of pi(e^x)^2 ?

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

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integral of pi(e^x)^2

\int π (e^{x})^{2} d x

\frac{π}{2} e^{2 x} + C

Solution steps

\int π (e^{x})^{2} d x

Take the constant out:

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

= π \cdot \int (e^{x})^{2} d x

Apply u-substitution

: \int u d u

= π \cdot \int u d u

Apply the Power Rule

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

= π \frac{u ^{2}}{2}

Substitute back

u = e^{x}

= π \frac{( e ^{x} ) ^{2}}{2}

Simplify

π \frac{( e ^{x} ) ^{2}}{2} : \frac{π}{2} e^{2 x}

= \frac{π}{2} e^{2 x}

Add a constant to the solution

= \frac{π}{2} e^{2 x} + C

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