"Integration by Substitution" is a method used in calculus to evaluate integrals. It involves substituting a variable in the integrand with another variable or expression in order to simplify the integral and make it easier to solve.
"Integration by Substitution" is used when the integrand contains a function that can be simplified by using a substitution. This method is especially useful when the integrand contains a composite function, where one function is nested inside another.
To perform "Integration by Substitution", follow these steps:
1. Identify the function that can be substituted in the integrand.
2. Choose an appropriate substitution, usually denoted by u.
3. Rewrite the integrand in terms of u.
4. Find the derivative of u and substitute it in the integral.
5. Solve the new integral.
6. Substitute back in the original variable to get the final answer.
The purpose of "Integration by Substitution" is to simplify an integral and make it easier to solve. It can also be used to evaluate integrals that cannot be solved using other methods.
Some common mistakes to avoid when using "Integration by Substitution" include:
- Choosing an inappropriate substitution
- Forgetting to substitute back in the original variable
- Misinterpreting the chain rule and not properly substituting the derivative
- Making algebraic errors when simplifying the integral
- Not checking the final answer using differentiation