How do i begin to solve this?
integralof dx/(cube root of 3x+4)
1/2 (3x+4)^2/3
take the derivative of that
that is actually the answer (that my book gives) how did you get that??
Use the substitution
t=3x+4, dt=3dx, or dx=dt/3
I=∫t^(-1/3)dt/3
The antiderivative can then be found by the power rule.
To evaluate the integral ∫ dx / ∛(3x + 4), you can use the substitution method or the u-substitution technique. Here's an explanation of the u-substitution method:
1. Let u = 3x + 4
- The goal is to simplify the integral by substituting u for a part of the function.
2. Find the derivative of u with respect to x: du/dx = 3
- By taking the derivative, we can determine the relationship between du and dx.
3. Solve for dx in terms of du:
- Rearrange the equation found in step 2: dx = du/3
4. Substitute the values of u and dx into the integral:
∫ dx / ∛(3x + 4) = ∫ (du/3) / ∛u
= (1/3)∫ du / u^(1/3)
5. Simplify the integral:
- Using the power rule for integration, the integral becomes:
(1/3)∫ u^(-1/3) du
6. Integrate:
- Apply the power rule to integrate: ∫ u^(-1/3) du = (3/2)u^(2/3) + C
(where C is the constant of integration)
7. Plug back the value of u:
=(3/2)(3x+4)^(2/3) + C
Therefore, the solution to the integral ∫ dx / ∛(3x + 4) is (3/2)(3x + 4)^(2/3) + C, where C is the constant of integration.