What is the integral of ((e^x))/(e^{2x)+16} ?

The integral of ((e^x))/(e^{2x)+16} is 1/4 arctan((e^x)/4)+C

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integral of ((e^x))/(e^{2x)+16}

\int \frac{( e ^{x} )}{e ^{2 x} + 16} d x

\frac{1}{4} arctan (\frac{e ^{x}}{4}) + C

Solution steps

\int \frac{e ^{x}}{e ^{2 x} + 16} d x

Apply u-substitution

= \int \frac{1}{u ^{2} + 16} d u

Apply Integral Substitution

= \int \frac{1}{4 ( v ^{2} + 1 )} d v

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= \frac{1}{4} arctan (v)

Substitute back

= \frac{1}{4} arctan (\frac{e ^{x}}{4})

Add a constant to the solution

= \frac{1}{4} arctan (\frac{e ^{x}}{4}) + C

What is the integral of ((e^x))/(e^{2x)+16} ?
The integral of ((e^x))/(e^{2x)+16} is 1/4 arctan((e^x)/4)+C