Answer:
To evaluate the integral ∫(x^3 + 4x^2)(x^2 - x + 4) dx, you can use the distributive property to expand the integrand and then integrate each term separately:
∫(x^3 + 4x^2)(x^2 - x + 4) dx = ∫x^5 - x^4 + 4x^3 + 4x^4 - 4x^3 + 16x^2 dx
Now, you can simplify the expression:
∫(x^5 - x^4 + 4x^3 + 4x^4 - 4x^3 + 16x^2) dx = ∫(x^5 + 3x^4 + 16x^2) dx
Next, integrate each term separately:
∫x^5 dx + ∫3x^4 dx + ∫16x^2 dx
Now, apply the power rule for integration:
∫x^5 dx = (1/6)x^6 + C1, where C1 is the constant of integration.
∫3x^4 dx = (3/5)x^5 + C2, where C2 is the constant of integration.
∫16x^2 dx = (16/3)x^3 + C3, where C3 is the constant of integration.
Now, combine the results:
(1/6)x^6 + C1 + (3/5)x^5 + C2 + (16/3)x^3 + C3
You can further simplify this expression by combining the constants of integration:
(1/6)x^6 + (3/5)x^5 + (16/3)x^3 + (C1 + C2 + C3)
So, the indefinite integral ∫(x^3 + 4x^2)(x^2 - x + 4) dx simplifies to:
(1/6)x^6 + (3/5)x^5 + (16/3)x^3 + C, where C = (C1 + C2 + C3) is the constant of integration.
Explanation: