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

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

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

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

\frac{e ^{2 x}}{( 1 - ln ( 2 ) ) \cdot 2 ^{1 + 2 x}} + C

Solution steps

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

Apply u-substitution

= \int (\frac{e}{2})^{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 (\frac{e}{2})^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

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

Substitute back

u = 2 x

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

Simplify

\frac{1}{2} \cdot \frac{( \frac{e}{2} ) ^{2 x}}{ln ( \frac{e}{2} )} : \frac{e ^{2 x}}{( 1 - ln ( 2 ) ) \cdot 2 ^{1 + 2 x}}

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

Add a constant to the solution

= \frac{e ^{2 x}}{( 1 - ln ( 2 ) ) \cdot 2 ^{1 + 2 x}} + C

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