What is the integral of (e^x)/(3+16e^{2x)} ?

The integral of (e^x)/(3+16e^{2x)} is 1/(4sqrt(3))arctan(4/(sqrt(3))e^x)+C

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integral of (e^x)/(3+16e^{2x)}

\int \frac{e ^{x}}{3 + 16 e ^{2 x}} d x

\frac{1}{4 3} arctan (\frac{4}{3} e^{x}) + C

Solution steps

\int \frac{e ^{x}}{3 + 16 e ^{2 x}} d x

Apply u-substitution

: \int \frac{1}{3 + 16 u ^{2}} d u

= \int \frac{1}{3 + 16 u ^{2}} d u

Apply Integral Substitution

= \int \frac{1}{4 3 ( 1 + v ^{2} )} d v

Take the constant out:

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

= \frac{1}{4 3} \cdot \int \frac{1}{1 + v ^{2}} d v

Use the common integral:

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

= \frac{1}{4 3} arctan (v)

Substitute back

= \frac{1}{4 3} arctan (\frac{4}{3} e^{x})

Add a constant to the solution

= \frac{1}{4 3} arctan (\frac{4}{3} e^{x}) + C

\int \frac{( 1 )}{( u ^{2} - 1 )} d u

f (x) = 5 x^{3}, at x = 2

\frac{\partial}{\partial x} (x y + cos (x))

\frac{d}{d x} f (x) = (2 x - 6)^{2} (x^{2} + x + 1)^{2}

\frac{d}{d x} (\frac{4}{sin ( x ) + cos ( x )})

What is the integral of (e^x)/(3+16e^{2x)} ?
The integral of (e^x)/(3+16e^{2x)} is 1/(4sqrt(3))arctan(4/(sqrt(3))e^x)+C