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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Apply Integral Substitution

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

Substitute back

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

Add a constant to the solution

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

\int_{0}^{1} x x^{2} + 2 d x

2 \frac{d y}{d x} + y = 0

y = x^{m x}

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

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

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