What is the integral of (30)/(30+e^x) ?

The integral of (30)/(30+e^x) is x-ln|e^x+30|+C

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integral of (30)/(30+e^x)

\int \frac{30}{30 + e ^{x}} d x

x - ln ∣ e^{x} + 30 ∣ + C

Solution steps

\int \frac{30}{30 + e ^{x}} d x

Take the constant out:

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

= 30 \cdot \int \frac{1}{30 + e ^{x}} d x

Apply u-substitution

= 30 \cdot \int \frac{1}{u ( 30 + u )} d u

Take the partial fraction of

\frac{1}{u ( 30 + u )} : \frac{1}{30 u} - \frac{1}{30 ( u + 30 )}

= 30 \cdot \int \frac{1}{30 u} - \frac{1}{30 ( u + 30 )} 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

= 30 (\int \frac{1}{30 u} d u - \int \frac{1}{30 ( u + 30 )} d u)

\int \frac{1}{30 u} d u = \frac{1}{30} ln ∣ u ∣

\int \frac{1}{30 ( u + 30 )} d u = \frac{1}{30} ln ∣ u + 30 ∣

= 30 (\frac{1}{30} ln ∣ u ∣ - \frac{1}{30} ln ∣ u + 30 ∣)

Substitute back

u = e^{x}

= 30 (\frac{1}{30} ln ∣ e^{x} ∣ - \frac{1}{30} ln ∣ e^{x} + 30 ∣)

Simplify

30 (\frac{1}{30} ln ∣ e^{x} ∣ - \frac{1}{30} ln ∣ e^{x} + 30 ∣) : x - ln ∣ e^{x} + 30 ∣

= x - ln ∣ e^{x} + 30 ∣

Add a constant to the solution

= x - ln ∣ e^{x} + 30 ∣ + C

y = 4 (6 - x^{2})^{5}

\int 4 m (9 m^{2} - 10 m) d m

ln (15 x)

f (x) = x^{12}

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

What is the integral of (30)/(30+e^x) ?
The integral of (30)/(30+e^x) is x-ln|e^x+30|+C