Question

Evaluate the indefinite integral∫(x2−4x 4)5(x−2)2dx.

A) x³ - 8x⁵ + C
B) (1/6)x³ - (4/7)x⁷ + C
C) (1/7)x⁶ - (2/3)x² + C
D) (1/8)x⁸ - (2/5)x³ + C

Answer 1

To evaluate this indefinite integral, we expand the expression, integrate each term separately, and combine the results.

Step-by-step explanation:

To evaluate the indefinite integral ∫(x^2 - 4x^4)5(x - 2)^2dx, we can expand the expression and then integrate each term separately.

First, let's simplify the expression inside the integral by multiplying the terms: (x^2 - 4x^4) * 5(x - 2)^2. Expanding this gives us: 5x^3 - 20x^5 + 20x^4 - 80x^6.

Now, we can integrate each term:

• The integral of 5x^3 is (5/4)x^4.
• The integral of -20x^5 is (-4/7)x^6.
• The integral of 20x^4 is (4/5)x^5.
• The integral of -80x^6 is (-16/7)x^7.

Combining these results, the indefinite integral of (x^2 - 4x^4)5(x - 2)^2dx is: C + (5/4)x^4 - (4/7)x^6 + (4/5)x^5 - (16/7)x^7, where C is the constant of integration.