What is the integral of (1.05)/(1+100e^{-0.285x)} ?

The integral of (1.05)/(1+100e^{-0.285x)} is -3.68421…(-0.285x-ln|100e^{-0.285x}+1|)+C

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integral of (1.05)/(1+100e^{-0.285x)}

\int \frac{1.05}{1 + 100 e ^{- 0.285 x}} d x

- 3.68421 \dots (- 0.285 x - ln 100 e^{- 0.285 x} + 1) + C

Solution steps

\int \frac{1.05}{1 + 100 e ^{- 0.285 x}} d x

Take the constant out:

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

= 1.05 \cdot \int \frac{1}{1 + 100 e ^{- 0.285 x}} d x

\frac{1}{1 + 100 e ^{- 0.285 x}} = - \frac{100 e ^{- 0.285 x}}{1 + 100 e ^{- 0.285 x}} + 1

= 1.05 \cdot \int - \frac{100 e ^{- 0.285 x}}{1 + 100 e ^{- 0.285 x}} + 1 d x

Apply u-substitution

: \int - \frac{1}{0.285 ( 1 + 100 e ^{u} )} d u

= 1.05 \cdot \int - \frac{1}{0.285 ( 1 + 100 e ^{u} )} d u

Take the constant out:

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

= 1.05 (- \frac{1}{0.285} \cdot \int \frac{1}{1 + 100 e ^{u}} d u)

Simplify

= 1.05 (- 3.50877 \dots \cdot \int \frac{1}{1 + 100 e ^{u}} d u)

Apply u-substitution

: \int \frac{1}{v ( 1 + 100 v )} d v

= 1.05 (- 3.50877 \dots \cdot \int \frac{1}{v ( 1 + 100 v )} d v)

Take the partial fraction of

\frac{1}{v ( 1 + 100 v )} : \frac{1}{v} - \frac{100}{100 v + 1}

= 1.05 (- 3.50877 \dots \cdot \int \frac{1}{v} - \frac{100}{100 v + 1} d v)

Apply the Sum Rule:

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

= 1.05 (- 3.50877 \dots (\int \frac{1}{v} d v - \int \frac{100}{100 v + 1} d v))

\int \frac{1}{v} d v = ln ∣ v ∣

\int \frac{100}{100 v + 1} d v = ln ∣ 100 v + 1 ∣

= 1.05 (- 3.50877 \dots (ln ∣ v ∣ - ln ∣ 100 v + 1 ∣))

Substitute back

= 1.05 (- 3.50877 \dots (ln e^{- 0.285 x} - ln 100 e^{- 0.285 x} + 1))

Simplify

1.05 (- 3.50877 \dots (ln e^{- 0.285 x} - ln 100 e^{- 0.285 x} + 1)) : - 3.68421 \dots (- 0.285 x - ln 100 e^{- 0.285 x} + 1)

= - 3.68421 \dots (- 0.285 x - ln 100 e^{- 0.285 x} + 1)

Add a constant to the solution

= - 3.68421 \dots (- 0.285 x - ln 100 e^{- 0.285 x} + 1) + C

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What is the integral of (1.05)/(1+100e^{-0.285x)} ?
The integral of (1.05)/(1+100e^{-0.285x)} is -3.68421…(-0.285x-ln|100e^{-0.285x}+1|)+C