FIND EACH INDEFINITE INTERGRAL BY THE SUBSTITUTION METHOD
X^3+X^2/3X^4+4X^3 DX
To find the indefinite integral of the given expression using the substitution method, follow these steps:
1. Let's start by identifying the substitution. Look for a portion of the expression that closely resembles the derivative of a function multiplied by a constant. In this case, the derivative of x^4 is 4x^3, so we can choose u = x^4 as our substitution.
2. Calculate the differential of u, du. Taking the derivative of u with respect to x, we have du/dx = 4x^3. Rearranging, we have du = 4x^3 dx.
3. Rewrite the original expression in terms of u using the substitution. Substituting u = x^4, we get x^3 + x^2 / (3x^4 + 4x^3) dx = (1 + 1/(3u)) * du.
4. Now, the integral of the expression becomes ∫ (1 + 1/(3u)) du.
5. Simplify the integral by expanding the term inside. ∫ 1 du + ∫ 1/(3u) du = ∫ 1 du + (1/3) ∫ (1/u) du.
6. Integrate each term separately. The integral of 1 du is simply u. The integral of 1/u du can be evaluated using the natural logarithm function: ∫ (1/u) du = ln|u|.
7. Putting it all together, we have ∫ (1 + 1/(3u)) du = u + (1/3) ln|u|.
8. Finally, substitute u back in terms of x: u = x^4. Thus, the indefinite integral of the given expression is x^4 + (1/3) ln|x^4| + C, where C is the constant of integration.
Therefore, the indefinite integral of x^3 + x^2 / (3x^4 + 4x^3) dx using the substitution method is x^4 + (1/3) ln|x^4| + C.