Question

Evaluate the following indefinite integrals:(a) ∫(-4x³ - x - √2) dx

(b) ∫(-4x³ - 2) dx

Answer 1

To evaluate the indefinite integrals, each term within the integral is integrated separately. The final answers include the antiderivative of each term plus a constant of integration, C.

Step-by-step explanation:

The task is to evaluate two indefinite integrals:

• For the first integral, ∑(-4x³ - x - √2) dx, we integrate term by term to find the antiderivative: -4x³ becomes -x⁴, -x becomes -x²/2, and -√2 becomes -√2x. Adding the constant of integration, C, gives us the final answer.
• The second integral, ∑(-4x³ - 2) dx, is similar: -4x³ integrates to -x⁴, -2 integrates to -2x, and we again include the constant of integration, C.

Thus, the answers are:

1. ∑(-4x³ - x - √2) dx = -x⁴/4 - x²/4 - √2x + C
2. ∑(-4x³ - 2) dx = -x⁴/4 - 2x + C