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

The integral of (3e^x+2)/(e^{2x)+e^x} is x-ln|e^x+1|-2/(e^x)+C

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

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

x - ln ∣ e^{x} + 1 ∣ - \frac{2}{e ^{x}} + C

Solution steps

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

Expand

\frac{3 e ^{x} + 2}{e ^{2 x} + e ^{x}} : \frac{3 e ^{x}}{e ^{2 x} + e ^{x}} + \frac{2}{e ^{2 x} + e ^{x}}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{3 e ^{x}}{e ^{2 x} + e ^{x}} d x = 3 (x - ln ∣ e^{x} + 1 ∣)

\int \frac{2}{e ^{2 x} + e ^{x}} d x = 2 (- x - \frac{1}{e ^{x}} + ln ∣ e^{x} + 1 ∣)

= 3 (x - ln ∣ e^{x} + 1 ∣) + 2 (- x - \frac{1}{e ^{x}} + ln ∣ e^{x} + 1 ∣)

Simplify

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

= x - ln ∣ e^{x} + 1 ∣ - \frac{2}{e ^{x}}

Add a constant to the solution

= x - ln ∣ e^{x} + 1 ∣ - \frac{2}{e ^{x}} + C

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