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 (5x3)4dx\int (5x - 3)^4 \, dx

We use substitution: Let u=5x3u = 5x - 3, then dudx=5dx=du5\frac{du}{dx} = 5 \Rightarrow dx = \frac{du}{5}.

The integral becomes: (5x3)4dx=u415du=15u4du\int (5x - 3)^4 dx = \int u^4 \cdot \frac{1}{5} du = \frac{1}{5} \int u^4 du

The integral of u4u^4 is: u55+C=(5x3)525+C\frac{u^5}{5} + C = \frac{(5x - 3)^5}{25} + C

Answer: (5x3)525+C\frac{(5x - 3)^5}{25} + C


2. Solve 4x(6x25)4dx\int 4x \cdot (6x^2 - 5)^4 \, dx

Again, use substitution: Let u=6x25u = 6x^2 - 5, so dudx=12xdx=du12x\frac{du}{dx} = 12x \Rightarrow dx = \frac{du}{12x}.

The xx in the numerator cancels with the xx in the denominator: 4x(6x25)4dx=412u4du=13u4du\int 4x (6x^2 - 5)^4 dx = \int \frac{4}{12} u^4 du = \frac{1}{3} \int u^4 du

The integral of u4u^4 is: u55+C=13(6x25)55+C\frac{u^5}{5} + C = \frac{1}{3} \cdot \frac{(6x^2 - 5)^5}{5} + C

Simplify: (6x25)515+C\frac{(6x^2 - 5)^5}{15} + C

Answer: (6x25)515+C\frac{(6x^2 - 5)^5}{15} + C


3. Solve 01(8x3+6x22x)dx\int_0^1 (8x^3 + 6x^2 - 2x) \, dx

We integrate term by term: 018x3dx=[2x4]01=2(1)42(0)4=2\int_0^1 8x^3 dx = \left[2x^4\right]_0^1 = 2(1)^4 - 2(0)^4 = 2 016x2dx=[2x3]01=2(1)32(0)3=2\int_0^1 6x^2 dx = \left[2x^3\right]_0^1 = 2(1)^3 - 2(0)^3 = 2 012xdx=[x2]01=(1)2(0)2=1\int_0^1 -2x dx = \left[-x^2\right]_0^1 = -(1)^2 - -(0)^2 = -1

Add them together: 2+21=32 + 2 - 1 = 3

Answer: 33


Let me know if you want further clarifications! Here are related questions you can consider:

  1. What are the steps for solving definite integrals using substitution?
  2. How do polynomial integrals work term-by-term?
  3. How do you verify if substitution is the correct method?
  4. How do definite integrals differ from indefinite integrals in terms of constants?
  5. 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