"Find the indefinite integral. ∫(5−ex)e2xdx" was answered by our community

waigaK

Answered question

2021-10-16

Find the indefinite integral.

\int \frac{(5 - e^{x})}{e^{2 x}} d x

comentezq

Skilled2021-10-17Added 106 answers

Step 1
We have find indefinite integral:

\int \frac{(5 - e^{x})}{e^{2 x}} d x

Separating each terms of the rational function of the integral,

\int \frac{(5 - e^{x})}{e^{2 x}} d x = \int (\frac{5}{e^{2 x}} - \frac{e^{x}}{e^{2 x}}) d x

= 5 \int e^{- 2 x} d x - \int e^{x - 2 x} d x

= 5 \int e^{- 2 x} d x - \int e^{- x} d x

Step 2
We know the formula of integration,

\int e^{x} d x = e^{x} + C

\int e^{- a x} d x = \frac{e^{- a x}}{- a} + C

So applying above formula,

5 \int e^{- 2 x} d x - \int e^{- x} d x = 5 (\frac{e^{- 2 x}}{- 2}) - (\frac{e^{- x}}{- 1}) + C

= - \frac{5}{2} e^{- 2 x} + e^{- x} + C

Where, C is an arbitrary constant.
Hence, value of integration is

- \frac{5}{2} e^{- 2 x} + e^{- x} + C

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