"Evaluate the integral. ∫(1+ex)2exdx" was answered by our experts

korporasidn

Answered question

2021-11-21

Evaluate the integral.

\int \frac{{(1 + e^{x})}^{2}}{e^{x}} d x

Tionant

Beginner2021-11-22Added 17 answers

Step 1
The given integral is

\int \frac{{(1 + e^{x})}^{2}}{e^{x}} d x

.
Step 2
Evaluate the given integral as shown below:

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

= \int \frac{1}{e^{x}} + 2 + e^{x} d x

= \int \frac{1}{e^{x}} d x + \int 2 d x + \int e^{x} d x

= - \frac{1}{e^{x}} + 2 x + e^{x} + C

Ancessitere

Beginner2021-11-23Added 17 answers

Step 1: Expand.

\int \frac{1 + 2 e^{x} + e^{2 x}}{e^{x}} d x

Step 2: Split fraction.

\int \frac{1}{e^{x}} + \frac{2 e^{x}}{e^{x}} + \frac{e^{2 x}}{e^{x}} d x

Step 3: Use Sum Rule:

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

\int \frac{1}{e^{x}} d x + \int 2 d x + \int e^{x} d x

Step 4: Use Integration by Substitution on

\int \frac{1}{e^{x}} d x

.
Let

u = \frac{1}{e^{x}}, d u = - \frac{1}{e^{x}} d x, t h e n d x = - e^{\ln \frac{1}{u}} d u

Step 5: Using u and du above, rewrite

\int \frac{1}{e^{x}} d x

\int u \times - e^{\ln \frac{1}{u}} d u

Step 6: Use this rule:

\int a d x = a x + C

.
-u
Step 7: Substitute

u = \frac{1}{e^{x}}

back into the original integral.

- \frac{1}{e^{x}}

Step 8: Rewrite the integral with the completed substitution.

- \frac{1}{e^{x}} + \int 2 d x + \int e^{x} d x

Step 9: Use this rule:

\int a d x = a x + C

- \frac{1}{e^{x}} + 2 x + \int e^{x} d x

Step 10: The integral of

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

- \frac{1}{e^{x}} + 2 x + e^{x}

Step 11: Add constant.

- \frac{1}{e^{x}} + 2 x + e^{x} + C

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