What is the integral from 0 to 2 of 3-(4e^{0.28t})/(e^{0.28t)+10.13} ?

The integral from 0 to 2 of 3-(4e^{0.28t})/(e^{0.28t)+10.13} is 5.06758…

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integral from 0 to 2 of 3-(4e^{0.28t})/(e^{0.28t)+10.13}

\int_{0}^{2} 3 - \frac{4 e ^{0.28 t}}{e ^{0.28 t} + 10.13} d t

5.06758 \dots

Solution steps

\int_{0}^{2} 3 - \frac{4 e ^{0.28 t}}{e ^{0.28 t} + 10.13} d t

Apply u-substitution

= \int_{0}^{0.56} \frac{- e ^{u} + 30.39}{0.28 ( e ^{u} + 10.13 )} d u

Take the constant out:

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

= \frac{1}{0.28} \cdot \int_{0}^{0.56} \frac{- e ^{u} + 30.39}{e ^{u} + 10.13} d u

Simplify

= 3.57142 \dots \cdot \int_{0}^{0.56} \frac{- e ^{u} + 30.39}{e ^{u} + 10.13} d u

Expand

\frac{- e ^{u} + 30.39}{e ^{u} + 10.13} : - \frac{e ^{u}}{e ^{u} + 10.13} + \frac{30.39}{e ^{u} + 10.13}

Apply the Sum Rule:

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

= 3.57142 \dots (- \int_{0}^{0.56} \frac{e ^{u}}{e ^{u} + 10.13} d u + \int_{0}^{0.56} \frac{30.39}{e ^{u} + 10.13} d u)

\int_{0}^{0.56} \frac{e ^{u}}{e ^{u} + 10.13} d u = 0.06526 \dots

\int_{0}^{0.56} \frac{30.39}{e ^{u} + 10.13} d u = 1.48419 \dots

= 3.57142 \dots (- 0.06526 \dots + 1.48419 \dots)

Simplify

= 5.06758 \dots

What is the integral from 0 to 2 of 3-(4e^{0.28t})/(e^{0.28t)+10.13} ?
The integral from 0 to 2 of 3-(4e^{0.28t})/(e^{0.28t)+10.13} is 5.06758…