Answered: "Evaluate the given integral. ∫5e−2xdx"

Jessica Scott

Answered question

2021-11-20

Evaluate the given integral.

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

Nicole Keller

Beginner2021-11-21Added 14 answers

Step 1
The given integral

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

Step 2
u=-2x

\frac{d u}{d x} = - 2

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

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

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

oces3y

Beginner2021-11-22Added 21 answers

Step 1: Use Constant Factor Rule:

\int c f (x) d x = c \int f (x) d x

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

Step 2: Use Integration by Substitution on

\int e^{- 2 x} d x

.
Let u=-2x, du=-2dx, then

d x = - \frac{1}{2} d u

Step 3: Using u and du above, rewrite

\int e^{- 2 x} d x

\int - \frac{e^{u}}{2} d u

Step 4: Use Constant Factor Rule:

\int c f (x) d x = c \int f (x) d x

- \frac{1}{2} \int e^{u} d u

Step 5: The integral of

e^{x} i s e^{x}

- \frac{e^{u}}{2}

Step 6: Substitute u=−2x back into the original integral.

- \frac{e^{- 2 x}}{2}

Step 7: Rewrite the integral with the completed substitution.

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

Step 8: Add constant.

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

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