What is the integral of (23)/(23+e^x) ?

The integral of (23)/(23+e^x) is x-ln|e^x+23|+C

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integral of (23)/(23+e^x)

\int \frac{23}{23 + e ^{x}} d x

x - ln ∣ e^{x} + 23 ∣ + C

Solution steps

\int \frac{23}{23 + e ^{x}} d x

Take the constant out:

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

= 23 \cdot \int \frac{1}{23 + e ^{x}} d x

Apply u-substitution

: \int \frac{1}{u ( 23 + u )} d u

= 23 \cdot \int \frac{1}{u ( 23 + u )} d u

Take the partial fraction of

\frac{1}{u ( 23 + u )} : \frac{1}{23 u} - \frac{1}{23 ( u + 23 )}

= 23 \cdot \int \frac{1}{23 u} - \frac{1}{23 ( u + 23 )} 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

= 23 (\int \frac{1}{23 u} d u - \int \frac{1}{23 ( u + 23 )} d u)

\int \frac{1}{23 u} d u = \frac{1}{23} ln ∣ u ∣

\int \frac{1}{23 ( u + 23 )} d u = \frac{1}{23} ln ∣ u + 23 ∣

= 23 (\frac{1}{23} ln ∣ u ∣ - \frac{1}{23} ln ∣ u + 23 ∣)

Substitute back

u = e^{x}

= 23 (\frac{1}{23} ln ∣ e^{x} ∣ - \frac{1}{23} ln ∣ e^{x} + 23 ∣)

Simplify

23 (\frac{1}{23} ln ∣ e^{x} ∣ - \frac{1}{23} ln ∣ e^{x} + 23 ∣) : x - ln ∣ e^{x} + 23 ∣

= x - ln ∣ e^{x} + 23 ∣

Add a constant to the solution

= x - ln ∣ e^{x} + 23 ∣ + C

\frac{\partial}{\partial x} (x y + ln (x + y))

\frac{d}{d x} (sin^{l n (x)} (3 x))

\int sin (a t) d t

\frac{d}{d x} (e^{3 x} + 10 e^{2 x})

t^{2} y^{^{'}} + 3 t y + 2 t^{3} = 0

What is the integral of (23)/(23+e^x) ?
The integral of (23)/(23+e^x) is x-ln|e^x+23|+C