Find the integral of (2x + 1)^3 dx. (2019)

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Question: Find the integral of (2x + 1)^3 dx. (2019)

Options:

Correct Answer: (1/4)(2x + 1)^4 + C

Solution:

Using substitution, the integral is (1/4)(2x + 1)^4 + C.

Find the integral of (2x + 1)^3 dx. (2019)

Find the integral of (2x + 1)^3 dx. (2019)

Step 1: Identify the integral you need to solve: ∫(2x + 1)³ dx.
Step 2: Use substitution. Let u = 2x + 1. This means that du/dx = 2, or du = 2 dx.
Step 3: Solve for dx in terms of du: dx = du/2.
Step 4: Substitute u and dx into the integral: ∫(u)³ (du/2).
Step 5: Factor out the 1/2: (1/2) ∫u³ du.
Step 6: Now, integrate u³: ∫u³ du = (1/4)u⁴ + C.
Step 7: Combine the results: (1/2) * (1/4)u⁴ + C = (1/8)u⁴ + C.
Step 8: Substitute back u = 2x + 1: (1/8)(2x + 1)⁴ + C.
Step 9: Simplify the expression: (1/4)(2x + 1)⁴ + C.

Integration by Substitution – This concept involves substituting a part of the integrand with a new variable to simplify the integration process.
Polynomial Integration – This concept focuses on integrating polynomial expressions, which often involves applying the power rule.