Math Problem Statement

Evaluate.

Integral from nothing to nothing left parenthesis x squared minus 7 x plus 2 right parenthesis font size decreased by 1 font size decreased by 1 dx∫x2−7x+2 dx

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Part 1

Integral from nothing to nothing left parenthesis x squared minus 7 x plus 2 right parenthesis font size decreased by 1 font size decreased by 1 dx∫x2−7x+2 dxequals=enter your response here

Solution

It looks like you want to evaluate the integral of the function x27x+2x^2 - 7x + 2 with respect to xx.

However, you didn't specify the limits of integration. I will proceed assuming this is an indefinite integral (i.e., no specific bounds are given).

The integral of x27x+2x^2 - 7x + 2 is computed term-by-term:

(x27x+2)dx=x2dx7xdx+2dx\int (x^2 - 7x + 2) \, dx = \int x^2 \, dx - \int 7x \, dx + \int 2 \, dx

We integrate each term:

  1. x2dx=x33\int x^2 \, dx = \frac{x^3}{3}
  2. 7xdx=7x22\int 7x \, dx = \frac{7x^2}{2}
  3. 2dx=2x\int 2 \, dx = 2x

So, the integral is:

x337x22+2x+C\frac{x^3}{3} - \frac{7x^2}{2} + 2x + C

where CC is the constant of integration.

If you need this evaluated with specific limits (from aa to bb), please let me know!


Would you like any further details or clarification on the solution?

Here are some related questions you might find useful:

  1. How do you evaluate a definite integral with limits of integration?
  2. What is the concept of the constant of integration in indefinite integrals?
  3. How can you check your result for definite integrals using the Fundamental Theorem of Calculus?
  4. How do you handle integration by substitution or parts for more complex functions?
  5. What happens when you apply integration to rational functions or trigonometric functions?

Tip: Always break the integral down term by term for polynomials; it simplifies the process significantly!

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Math Problem Analysis

Mathematical Concepts

Calculus
Indefinite Integral
Polynomial Integration

Formulas

∫x² dx = x³/3
∫ax dx = ax²/2
∫c dx = cx

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 10-12