𝑦 =

4

3−𝑥 find definite integral, lower limit 0 upper limit 2

To find the definite integral of \( y = \frac{4}{3-x} \) with the lower limit 0 and the upper limit 2, we follow these steps:

1. Integrate the function.
2. Evaluate the integral at the upper and lower limits and find the difference.

### Step 1: Integrate the function

We need to integrate \( \frac{4}{3-x} \) with respect to \( x \):

\[ \int \frac{4}{3-x} \, dx \]

To integrate this, we use the substitution method. Let \( u = 3 - x \). Then, \( du = -dx \) or \( dx = -du \).

Now, rewrite the integral in terms of \( u \):

\[ \int \frac{4}{u} \cdot (-du) = -4 \int \frac{1}{u} \, du \]

The integral of \( \frac{1}{u} \) is \( \ln|u| \):

\[ -4 \ln|u| + C \]

Substitute back \( u = 3 - x \):

\[ -4 \ln|3 - x| + C \]

### Step 2: Evaluate the integral at the upper and lower limits

Now, we evaluate the definite integral from 0 to 2:

\[ \left[ -4 \ln|3 - x| \right]_{0}^{2} \]

First, evaluate at the upper limit \( x = 2 \):

\[ -4 \ln|3 - 2| = -4 \ln(1) = -4 \cdot 0 = 0 \]

Then, evaluate at the lower limit \( x = 0 \):

\[ -4 \ln|3 - 0| = -4 \ln(3) \]

Now, find the difference between the evaluations at the upper and lower limits:

\[ 0 - (-4 \ln(3)) = 4 \ln(3) \]

### Conclusion

The definite integral of \( y = \frac{4}{3-x} \) from 0 to 2 is:

\[ \int_{0}^{2} \frac{4}{3-x} \, dx = 4 \ln(3) \]

Would you like more details or have any questions about this solution?

Here are 8 related questions for further practice:
1. Find the definite integral of \( y = \frac{5}{2-x} \) from 1 to 2.
2. Evaluate \( \int_{0}^{3} \frac{6}{4-x} \, dx \).
3. Compute the definite integral of \( y = \frac{3}{5-x} \) from 1 to 4.
4. Determine the area under the curve \( y = \frac{8}{1-x} \) between \( x = -1 \) and \( x = 0 \).
5. Find \( \int_{-1}^{1} \frac{7}{3+x} \, dx \).
6. Evaluate \( \int_{1}^{2} \frac{2}{4-x} \, dx \).
7. Compute the definite integral of \( y = \frac{5}{6-x} \) from 0 to 3.
8. Determine \( \int_{0}^{4} \frac{4}{5-x} \, dx \).

**Tip:** When dealing with integrals of the form \( \int \frac{1}{a-x} \, dx \), consider using substitution to simplify the integration process.

Learn how to compute the definite integral of y = 4/(3-x) from 0 to 2 using the substitution method. Detailed step-by-step solution provided.

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