Math Problem Statement

Solve the integrals provided in the image. Problems include polynomial expansions and definite integrals.

Solution

The uploaded image contains several integral problems. I'll extract and solve each of them step by step. Here's the breakdown of the integrals:

Problem Identification:

  1. Problem (a):
    (7x26x)(7x3)dx\int (7x^2 - 6x) (7x - 3) \, dx

  2. Problem (b):
    (4x36x27x10)dx\int (-4x^3 - 6x^2 - 7x - 10) \, dx

  3. Problem (c):
    (2x5)(2x6)dx\int (2x - 5)(2x - 6) \, dx

  4. Problem (d):
    (4x43)dx\int (-4x^4 - 3) \, dx

Let me solve each problem step by step. sympy import symbols, integrate, expand

Define the variable

x = symbols('x')

Define each integral problem

Problem (a)

expr_a = (7x**2 - 6x) * (7*x - 3) integral_a = integrate(expr_a, x)

Problem (b)

expr_b = -4x**3 - 6x**2 - 7*x - 10 integral_b = integrate(expr_b, x)

Problem (c)

expr_c = (2x - 5) * (2x - 6) integral_c = integrate(expr_c, x)

Problem (d)

expr_d = -4*x**4 - 3 integral_d = integrate(expr_d, x)

Expanding expressions for clarity

expanded_a = expand(expr_a) expanded_c = expand(expr_c)

integral_a, expanded_a, integral_b, integral_c, expanded_c, integral_dHere are the solutions to the integrals:

Problem (a):

(7x26x)(7x3)dx\int (7x^2 - 6x)(7x - 3) \, dx Expanded form:
49x363x2+18x49x^3 - 63x^2 + 18x
Integral:
49x4421x3+9x2+C\frac{49x^4}{4} - 21x^3 + 9x^2 + C


Problem (b):

(4x36x27x10)dx\int (-4x^3 - 6x^2 - 7x - 10) \, dx
Integral:
x442x37x2210x+C-\frac{x^4}{4} - 2x^3 - \frac{7x^2}{2} - 10x + C


Problem (c):

(2x5)(2x6)dx\int (2x - 5)(2x - 6) \, dx
Expanded form:
4x222x+304x^2 - 22x + 30
Integral:
4x3311x2+30x+C\frac{4x^3}{3} - 11x^2 + 30x + C


Problem (d):

(4x43)dx\int (-4x^4 - 3) \, dx
Integral:
4x553x+C-\frac{4x^5}{5} - 3x + C

Would you like further details on any step? Here are five follow-up questions to deepen understanding:

  1. How do we handle products of polynomials during integration?
  2. Why do we need to expand the product of terms before integrating?
  3. What is the significance of the constant of integration CC?
  4. How can we verify the correctness of these integrals?
  5. What is the process for integrating higher-degree polynomials?

Tip:

Always simplify and expand polynomial expressions before integration to make the process straightforward and reduce errors.

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Math Problem Analysis

Mathematical Concepts

Integral Calculus
Polynomial Integration
Algebraic Manipulation

Formulas

Power rule for integration: ∫x^n dx = (x^(n+1))/(n+1) + C (for n ≠ -1)
Linear expansion of polynomials: (a + b)(c + d) = ac + ad + bc + bd

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12 or early college-level calculus