What is the integral of (26)/(26+e^x) ?

The integral of (26)/(26+e^x) is x-ln|e^x+26|+C

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integral of (26)/(26+e^x)

\int \frac{26}{26 + e ^{x}} d x

x - ln ∣ e^{x} + 26 ∣ + C

Solution steps

\int \frac{26}{26 + e ^{x}} d x

Take the constant out:

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

= 26 \cdot \int \frac{1}{26 + e ^{x}} d x

Apply u-substitution

: \int \frac{1}{u ( 26 + u )} d u

= 26 \cdot \int \frac{1}{u ( 26 + u )} d u

Take the partial fraction of

\frac{1}{u ( 26 + u )} : \frac{1}{26 u} - \frac{1}{26 ( u + 26 )}

= 26 \cdot \int \frac{1}{26 u} - \frac{1}{26 ( u + 26 )} d u

Apply the Sum Rule:

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

= 26 (\int \frac{1}{26 u} d u - \int \frac{1}{26 ( u + 26 )} d u)

\int \frac{1}{26 u} d u = \frac{1}{26} ln ∣ u ∣

\int \frac{1}{26 ( u + 26 )} d u = \frac{1}{26} ln ∣ u + 26 ∣

= 26 (\frac{1}{26} ln ∣ u ∣ - \frac{1}{26} ln ∣ u + 26 ∣)

Substitute back

u = e^{x}

= 26 (\frac{1}{26} ln ∣ e^{x} ∣ - \frac{1}{26} ln ∣ e^{x} + 26 ∣)

Simplify

26 (\frac{1}{26} ln ∣ e^{x} ∣ - \frac{1}{26} ln ∣ e^{x} + 26 ∣) : x - ln ∣ e^{x} + 26 ∣

= x - ln ∣ e^{x} + 26 ∣

Add a constant to the solution

= x - ln ∣ e^{x} + 26 ∣ + C

\frac{x}{x + 6}

\int_{6}^{8} \frac{24}{( x - 6 ) ^{3}} d x

f (x) = 5 x, (25, 25)

\frac{d ^{2} y}{d x ^{2}} + 6 \frac{d y}{d x} + 9 y = 0

y^{^{'}} + \frac{1}{165} y = 0.48

What is the integral of (26)/(26+e^x) ?
The integral of (26)/(26+e^x) is x-ln|e^x+26|+C