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

The integral of ((e^x)^2)/2 is 1/4 e^{2x}+C

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integral of ((e^x)^2)/2

integral

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

\frac{1}{4} e^{2 x} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int u d u

= \frac{1}{2} \cdot \int u d u

Apply the Power Rule

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

= \frac{1}{2} \cdot \frac{u ^{2}}{2}

Substitute back

u = e^{x}

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

Simplify

\frac{1}{2} \cdot \frac{( e ^{x} ) ^{2}}{2} : \frac{1}{4} e^{2 x}

= \frac{1}{4} e^{2 x}

Add a constant to the solution

= \frac{1}{4} e^{2 x} + C

\frac{1}{( x ^{2} - 1 ) ^{3}}

(\frac{x + 1}{x})^{2}

\frac{sec ^{2} ( x )}{tan ( x )}

\frac{sin ^{2} ( x )}{cos ( x )}

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

What is the integral of ((e^x)^2)/2 ?
The integral of ((e^x)^2)/2 is 1/4 e^{2x}+C