What is the integral of (7e^{7x})/(1+e^{7x)} ?

The integral of (7e^{7x})/(1+e^{7x)} is ln|1+e^{7x}|+C

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integral of (7e^{7x})/(1+e^{7x)}

\int \frac{7 e ^{7 x}}{1 + e ^{7 x}} d x

ln 1 + e^{7 x} + C

Solution steps

\int \frac{7 e ^{7 x}}{1 + e ^{7 x}} d x

Take the constant out:

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

= 7 \cdot \int \frac{e ^{7 x}}{1 + e ^{7 x}} d x

Apply u-substitution

: \int \frac{1}{7 u} d u

= 7 \cdot \int \frac{1}{7 u} d u

Take the constant out:

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

= 7 \cdot \frac{1}{7} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 7 \cdot \frac{1}{7} ln ∣ u ∣

Substitute back

u = 1 + e^{7 x}

= 7 \cdot \frac{1}{7} ln 1 + e^{7 x}

Simplify

7 \cdot \frac{1}{7} ln 1 + e^{7 x} : ln 1 + e^{7 x}

= ln 1 + e^{7 x}

Add a constant to the solution

= ln 1 + e^{7 x} + C

x \to 0 lim (\frac{tan ( 12 x )}{sin ( 4 x )})

\frac{\partial}{\partial x} (y e^{x} - x^{2} y^{- 3})

y = \frac{4 x}{x ^{2} + 1}, (0, 0)

x \to - 1 lim (x^{2} - 2 x + 3)

\frac{d}{d x} (- \frac{cos ^{4} ( x )}{4})

What is the integral of (7e^{7x})/(1+e^{7x)} ?
The integral of (7e^{7x})/(1+e^{7x)} is ln|1+e^{7x}|+C