FIND EACH INDEFINITE INTERGRAL BY THE SUBSTITUTION METHOD
�çX^3+X^2/3X^4+4X^3 DX
To find the indefinite integral of the expression X^3+X^2/3X^4+4X^3 with respect to X, we can use the substitution method.
1. Let's start by rewriting the expression as:
∫ (X^3 + X^2) / (3X^4 + 4X^3) dX
2. We can then observe that the denominator can be factored as:
∫ (X^3 + X^2) / (X^3 (3X + 4)) dX
3. Now, we can select a substitution value. Let's choose u = X^3.
Therefore, du/dX = 3X^2, and solving for dX, we have dX = du / (3X^2).
4. Next, we substitute these values into the original expression:
∫ (X^3 + X^2) / (X^3 (3X + 4)) dX = ∫ (u + X^2) / (u (3X + 4)) (du / 3X^2)
5. Now, we can simplify the expression further:
∫ (1/u + X^2/u) (du / 3X^2 (3X + 4))
6. Combining like terms:
∫ (1/u + X^2/u) (du / 3X^4 + 4X^3)
7. Next, we can break down the integral into two separate integrals:
∫ (1/u) (du / 3X^4 + 4X^3) + ∫ (X^2/u) (du / 3X^4 + 4X^3)
8. Simplifying further:
(1/3) ∫ du/u + (1/3) ∫ (X^2/u) du/(X^3 + 4X^2)
9. The first integral, (1/3) ∫ du/u, simplifies to:
(1/3) ln|u| + C
10. The second integral, (1/3) ∫ (X^2/u) du/(X^3 + 4X^2), can be handled using a substitution.
11. Let's consider v = X^3 + 4X^2. Therefore, dv/dX = 3X^2 + 8X, and solving for dX, we get dX = dv/(3X^2 + 8X).
12. Now, we can substitute these values into the second integral:
(1/3) ∫ (X^2/u) du/(X^3 + 4X^2) = (1/3) ∫ ((X^2/u) (dv/(3X^2 + 8X))
13. Simplifying further:
(1/3) ∫ (X^2/u) dv/(3X^2 + 8X)
14. We can rearrange the expression to separate the variables:
(1/3) ∫ (X^2/u) / (3X^2 + 8X) dv
15. Now, we perform the integral of the second term using partial fractions or any appropriate method for the new expression. After integrating, we substitute back the value of u and X.
16. Once we have the result of the second integral in terms of v and u, we substitute back the value of v in terms of X.
17. Finally, we add the results from steps 9 and 16 to get the complete solution.
Please note that step 15 involves solving the integral of a rational function, which can be quite involved and depend on the specific values of X^3+X^2/3X^4+4X^3.