Indefinite Integrals: Integration by Substitution/Parts An indefinite integral represents the area under the curve of a function. In some cases, finding...
An indefinite integral represents the area under the curve of a function. In some cases, finding the exact area can be challenging or impossible. Fortunately, there are techniques like integration by substitution and parts to help us solve these integrals.
Integration by Substitution:
The inner function is the function you integrate with respect to.
The outer function is the function that integrates with respect to.
Replace the original variable in the integral with a new variable in the outer function.
This creates an equivalent integral in the outer function.
Apply the appropriate integration formula to the outer function.
Evaluate the integral and substitute back the original variable to find the final answer.
Parts:
Choose the inner function and the outer function based on their derivatives and limits of integration.
Evaluate the integrals and apply the formula to find the final answer.
Examples:
1. Evaluate the indefinite integral of (\int (x^2) dx):
Substitution: (u = x^2), (dv = dx)
where (C) is the constant of integration.
2. Evaluate the indefinite integral of (\int \frac{1}{x} dx):
Parts: (u = x, dv = \frac{1}{x} dx)
where (C) is the constant of integration.
These are just basic examples. Many more complex and interesting problems can be solved using these techniques