What is the integral of (21)/(21+e^x) ?

The integral of (21)/(21+e^x) is x-ln|e^x+21|+C

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integral of (21)/(21+e^x)

\int \frac{21}{21 + e ^{x}} d x

x - ln ∣ e^{x} + 21 ∣ + C

Solution steps

\int \frac{21}{21 + e ^{x}} d x

Take the constant out:

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

= 21 \cdot \int \frac{1}{21 + e ^{x}} d x

Apply u-substitution

: \int \frac{1}{u ( 21 + u )} d u

= 21 \cdot \int \frac{1}{u ( 21 + u )} d u

Take the partial fraction of

\frac{1}{u ( 21 + u )} : \frac{1}{21 u} - \frac{1}{21 ( u + 21 )}

= 21 \cdot \int \frac{1}{21 u} - \frac{1}{21 ( u + 21 )} 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

= 21 (\int \frac{1}{21 u} d u - \int \frac{1}{21 ( u + 21 )} d u)

\int \frac{1}{21 u} d u = \frac{1}{21} ln ∣ u ∣

\int \frac{1}{21 ( u + 21 )} d u = \frac{1}{21} ln ∣ u + 21 ∣

= 21 (\frac{1}{21} ln ∣ u ∣ - \frac{1}{21} ln ∣ u + 21 ∣)

Substitute back

u = e^{x}

= 21 (\frac{1}{21} ln ∣ e^{x} ∣ - \frac{1}{21} ln ∣ e^{x} + 21 ∣)

Simplify

21 (\frac{1}{21} ln ∣ e^{x} ∣ - \frac{1}{21} ln ∣ e^{x} + 21 ∣) : x - ln ∣ e^{x} + 21 ∣

= x - ln ∣ e^{x} + 21 ∣

Add a constant to the solution

= x - ln ∣ e^{x} + 21 ∣ + C

y^{^{'}} = \frac{3 x ^{2} + 2 x + 1}{y - 2}

\int_{0}^{2} \frac{x}{x + x ^{2}} d x

\int \frac{9}{x ^{2}} d x

\frac{1}{10 s}

y^{^{'}} = 3 y (5 - y)

What is the integral of (21)/(21+e^x) ?
The integral of (21)/(21+e^x) is x-ln|e^x+21|+C