3.3 Q-5

Question Statement

Evaluate the following integral:

$$\int \frac{e^{x}}{e^{x}+3} , dx$$

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Background and Explanation

This integral involves the exponential function in both the numerator and the denominator. We can simplify this expression using the substitution method. By substituting the denominator expression with a new variable, we can convert the integral into a simpler form.

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Solution

1. Substitute to simplify:
   Let the substitution be:

$$u = e^x + 3$$

Then, differentiate both sides with respect to $x$:

$$\frac{du}{dx} = e^x \quad \text{or} \quad du = e^x , dx$$

2. Rewrite the integral:
   Substituting $u = e^x + 3$ and $du = e^x , dx$ into the original integral:

$$\int \frac{e^x}{e^x + 3} , dx = \int \frac{1}{u} , du$$

3. Solve the integral:
   The integral of $\frac{1}{u}$ is a standard form:

$$\int \frac{1}{u} , du = \ln |u| + C$$

4. Substitute back the value of $u$:
   Now, substitute back $u = e^x + 3$:

$$\ln |e^x + 3| + C$$

Since $e^x + 3 > 0$ for all real $x$, we can drop the absolute value:

$$\ln (e^x + 3) + C$$

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Key Formulas or Methods Used

• Substitution:
  We used the substitution $u = e^x + 3$ to simplify the integral.

• Standard Integral:
  The integral of $\frac{1}{u}$ is:

$$\int \frac{1}{u} , du = \ln |u| + C$$

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Summary of Steps

1. Let $u = e^x + 3$ and differentiate to get $du = e^x , dx$.
2. Rewrite the integral as $\int \frac{1}{u} , du$.
3. Integrate to get $\ln |u| + C$.
4. Substitute back $u = e^x + 3$ to get the final answer:
   $\ln (e^x + 3) + C$