Math Problem Statement
Solve the integrals: 1) ∫ (5x - 3)^4 dx, 2) ∫ 4x * (6x^2 - 5)^4 dx, 3) ∫_0^1 (8x^3 + 6x^2 - 2x) dx.
Solution
Let’s solve the problems step by step.
1. Solve
We use substitution: Let , then .
The integral becomes:
The integral of is:
Answer:
2. Solve
Again, use substitution: Let , so .
The in the numerator cancels with the in the denominator:
The integral of is:
Simplify:
Answer:
3. Solve
We integrate term by term:
Add them together:
Answer:
Let me know if you want further clarifications! Here are related questions you can consider:
- What are the steps for solving definite integrals using substitution?
- How do polynomial integrals work term-by-term?
- How do you verify if substitution is the correct method?
- How do definite integrals differ from indefinite integrals in terms of constants?
- What happens when a substitution introduces a new integration bound?
Tip: Always check your substitution carefully to
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Math Problem Analysis
Mathematical Concepts
Integration
Substitution Method
Definite Integrals
Polynomial Integration
Formulas
∫ (a*x + b)^n dx = (1/n+1) * (a*x + b)^(n+1) + C
∫ u^n du = (u^(n+1))/(n+1) + C
∫_a^b f(x) dx = F(b) - F(a)
Theorems
Substitution Rule for Integration
Definite Integral Theorem
Suitable Grade Level
Grades 11-12