Kerala KEAM 5 June 2024 Solved Paper

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), $\frac{1}{2} (x^4 - 8x^2 + 16x)^2 + C$. Quick Tip: 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.