FIND EACH INDEFINITE INTERGRAL BY THE SUBSTITUTION METHOD
X^3+X^2/3X^4+4X^3 DX
To find each indefinite integral by the substitution method, we need to follow a systematic process.
Step 1: Identify the substitution. Let's set u equal to the expression inside the brackets. In this case, u = x^4 + 4x^3.
Step 2: Compute du/dx. Take the derivative of u with respect to x. In this case, the derivative of u with respect to x is du/dx = 4x^3 + 12x^2.
Step 3: Solve for dx. Rearrange the previous expression to isolate dx.
dx = du / (4x^3 + 12x^2)
Step 4: Substitute the values in the integral. Replace the expression inside the integral with u and dx with the derived expression.
∫ (x^3 + x^2) / (3x^4 + 4x^3) dx becomes:
∫ (1/((3x^4 + 4x^3)((4x^3 + 12x^2)))) du
Step 5: Simplify the integral. Expand and simplify the denominator.
∫ (1/(12x^6 + 52x^5 + 48x^4 + 144x^3)) du
Step 6: Separate the fractions and write the integral in partial fraction form. You may need to factorize the denominator and find the values of the constants.
Step 7: Integrate each term. Once the integral is written in partial fractions, integrate each term separately.
Step 8: Substitute back the original variable. Replace u with the original expression you substituted for at the beginning.
By following these steps, you will be able to find the indefinite integral by the substitution method.