Intergrate (x²+3x+1)(x³+7x²+2) out of x² dx?

To integrate (x²+3x+1)(x³+7x²+2) with respect to x², we can expand the expression and then integrate each term separately.

First, let's expand the expression:
(x²+3x+1)(x³+7x²+2) = x²(x³) + x²(7x²) + x²(2) + 3x(x³) + 3x(7x²) + 3x(2) + 1(x³) + 1(7x²) + 1(2)
= x^5 + 7x^4 + 2x² + 3x^4 + 21x³ + 6x + x³ + 7x² + 2

Now, we integrate each term with respect to x²:
∫(x^5 + 7x^4 + 2x² + 3x^4 + 21x³ + 6x + x³ + 7x² + 2) dx²
= (1/3)x^6 + (7/5)x^5 + (2/3)x^3 + (3/5)x^5 + (21/4)x^4 + 6x + (1/4)x^4 + 7x^3 + 2x + C

Therefore, the integrated expression is:
(1/3)x^6 + (22/5)x^5 + (2/3)x^3 + (25/4)x^4 + 7x^3 + 8x + C