Question

The integral \( \int (x^4 - 8x^2 + 16x)(4x^3 - 16x + 16) \, dx \) is:

• A. \( x^4 + 4x^3 - 8x^2 + 16x + 7 + C \)
• B. \( \frac{1}{2} (x^4 - 8x^2 + 16x + 7)^2 + C \)
• C. \( \frac{1}{2} (x^4 - 8x^2 + 16x)^2 + C \)
• D. \( \frac{1}{2} (x^4 - 8x^2 + 7)^2 + C \)
• E. \( \frac{1}{4} (x^4 - 8x^2 + 16x)^2 + C \)

Hint:

When faced with an integral involving products of polynomials, try to expand the terms first and identify any patterns that can simplify the expression, such as perfect squares.

Answer 1

The Correct Option is C

We are given the integral: \[ I = \int (x^4 - 8x^2 + 16x)(4x^3 - 16x + 16) \, dx \] Notice that the integrand is the product of two polynomials. 
We can simplify the multiplication first. Expand the terms: \[ (x^4 - 8x^2 + 16x)(4x^3 - 16x + 16) \] First, distribute \( (x^4 - 8x^2 + 16x) \) with each term of \( (4x^3 - 16x + 16) \): \[ = x^4(4x^3 - 16x + 16) - 8x^2(4x^3 - 16x + 16) + 16x(4x^3 - 16x + 16) \] \[ = 4x^7 - 16x^5 + 16x^4 - 32x^5 + 128x^3 - 128x^2 + 64x^4 - 256x^2 + 256x \] Now, collect like terms: \[ = 4x^7 - 48x^5 + 80x^4 + 128x^3 - 384x^2 + 256x \] Now, observe that this expression can be simplified further, but we notice the form of the answer choices. Since the integral involves a perfect square and matches the pattern of the answer choices, we recognize that: \[ \int (x^4 - 8x^2 + 16x)(4x^3 - 16x + 16) \, dx = \frac{1}{2} \left( x^4 - 8x^2 + 16x \right)^2 + C \] Thus, the integral simplifies to the form given in option (C). 
Thus, the correct answer is option (C), \( \frac{1}{2} (x^4 - 8x^2 + 16x)^2 + C \).