What is the definite integral of the function y = x^2+4x-3 precisely AT x = 5? Write your answer to the nearest 0.1.

To find the definite integral of the function \( y = x^2 + 4x - 3 \) at \( x = 5 \), we calculate the integral of the function from 0 to 5. The definite integral of a function \( f(x) \) from \( a \) to \( b \) is:

\[
\int_a^b f(x) \, dx
\]

Here, \( f(x) = x^2 + 4x - 3 \), and we want to integrate from 0 to 5.

### Step 1: Find the indefinite integral
The indefinite integral of \( f(x) = x^2 + 4x - 3 \) is:

\[
\int (x^2 + 4x - 3) \, dx = \frac{x^3}{3} + 2x^2 - 3x + C
\]

### Step 2: Apply the limits
Now, we evaluate this at the limits 0 and 5:

\[
\left[ \frac{x^3}{3} + 2x^2 - 3x \right]_0^5
\]

At \( x = 5 \):

\[
\frac{5^3}{3} + 2(5^2) - 3(5) = \frac{125}{3} + 2(25) - 15 = 41.67 + 50 - 15 = 76.67
\]

At \( x = 0 \):

\[
\frac{0^3}{3} + 2(0^2) - 3(0) = 0
\]

### Step 3: Calculate the definite integral
Now, subtract the result at \( x = 0 \) from the result at \( x = 5 \):

\[
76.67 - 0 = 76.67
\]

### Final Answer
The value of the definite integral from 0 to 5 of \( x^2 + 4x - 3 \) is approximately \( 76.7 \) to the nearest 0.1.

Would you like more details on any step or have any questions?

Here are five related questions to explore:
1. What is the definite integral from 1 to 5 of \( x^2 + 4x - 3 \)?
2. How do you find the area under a curve between two points?
3. How does changing the limits of integration affect the result?
4. What is the difference between definite and indefinite integrals?
5. How would the integral change if the function was \( x^2 + 4x + 3 \) instead?

**Tip**: The definite integral represents the area under the curve between two points on the x-axis.

What is the definite integral of the function y = x^2 + 4x - 3 precisely at x = 5? Write your answer to the nearest 0.1.

Explore the step-by-step solution to finding the definite integral of the function y = x^2 + 4x - 3 from 0 to 5. This problem involves key calculus concepts including definite and indefinite integrals and the application of the Fundamental Theorem of Calculus. Suitable for students in Grades 10-12.

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