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

The integral of (e^x)/(e^{2x)+e^x+2} is 2/(sqrt(7))arctan(1/(sqrt(7))(2e^x+1))+C

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

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

\frac{2}{7} arctan (\frac{1}{7} (2 e^{x} + 1)) + C

Solution steps

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

Apply u-substitution

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

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

Complete the square

u^{2} + u + 2 : (u + \frac{1}{2})^{2} + \frac{7}{4}

= \int \frac{1}{( u + \frac{1}{2} ) ^{2} + \frac{7}{4}} d u

Apply u-substitution

: \int \frac{4}{4 v ^{2} + 7} d v

= \int \frac{4}{4 v ^{2} + 7} d v

Take the constant out:

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

= 4 \cdot \int \frac{1}{4 v ^{2} + 7} d v

Apply Integral Substitution

= 4 \cdot \int \frac{1}{2 7 ( w ^{2} + 1 )} d w

Take the constant out:

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

= 4 \cdot \frac{1}{2 7} \cdot \int \frac{1}{w ^{2} + 1} d w

Use the common integral:

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

= 4 \cdot \frac{1}{2 7} arctan (w)

Substitute back

= 4 \cdot \frac{1}{2 7} arctan (\frac{2}{7} (e^{x} + \frac{1}{2}))

Simplify

4 \cdot \frac{1}{2 7} arctan (\frac{2}{7} (e^{x} + \frac{1}{2})) : \frac{2}{7} arctan (\frac{1}{7} (2 e^{x} + 1))

= \frac{2}{7} arctan (\frac{1}{7} (2 e^{x} + 1))

Add a constant to the solution

= \frac{2}{7} arctan (\frac{1}{7} (2 e^{x} + 1)) + C

\frac{d}{d x} ((e^{x^{2}} + 4)^{2} (4 x)^{3 x})

x \to 0 + lim (x^{6 s i n (x)})

\int_{0}^{\frac{π}{2}} \frac{cos ( t )}{1 + sin ^{2} ( t )} d t

\int \frac{10}{x ^{2} x ^{2} + 16} d x

x \to 0 lim (- (e^{x}))

What is the integral of (e^x)/(e^{2x)+e^x+2} ?
The integral of (e^x)/(e^{2x)+e^x+2} is 2/(sqrt(7))arctan(1/(sqrt(7))(2e^x+1))+C