Evaluate∫(2x + 1)1/3dx

To evaluate the integral ∫(2x + 1)^(1/3) dx, we can use the method of substitution. Here are the steps:

1. Let u = 2x + 1. Then, du = 2dx.

2. We need to express dx in terms of du. So, dx = du/2.

3. Now, substitute u and dx in the original integral. We get ∫u^(1/3) * (du/2).

4. Take the constant 1/2 out of the integral. We get (1/2) ∫u^(1/3) du.

5. Now, we can integrate u^(1/3). The integral of u^n with respect to u is (u^(n+1))/(n+1). So, the integral of u^(1/3) with respect to u is (3/4)u^(4/3).

6. Substitute this back into the integral. We get (1/2) * (3/4)u^(4/3) = (3/8)u^(4/3).

7. Finally, substitute u = 2x + 1 back into the integral. We get (3/8)(2x + 1)^(4/3).

So, the integral ∫(2x + 1)^(1/3) dx = (3/8)(2x + 1)^(4/3) + C, where C is the constant of integration.

To evaluate the integral ∫(2x + 1)^(1/3) dx, we can use the method of substitution.

Step 1: Let u = 2x + 1. Then, du = 2dx.

Step 2: We need to express dx in terms of du. So, dx = du/2.

Step 3: Substitute u and dx in the original integral. We get ∫u^(1/3) * (du/2).

Step 4: The constant 1/2 can be taken out of the integral. So, we get (1/2)∫u^(1/3) du.

Step 5: Now, we can integrate u^(1/3) with respect to u. The integral of u^n with respect to u is (u^(n+1))/(n+1). So, the integral of u^(1/3) with respect to u is (3/4)u^(4/3).

Step 6: Substitute back u = 2x + 1. So, the integral becomes (1/2)(3/4)(2x + 1)^(4/3).

Step 7: Simplify the expression to get the final answer. The final answer is (3/8)*(2x + 1)^(4/3) + C, where C is the constant of integration.