So, why do you have 1/7 * 1/7? Just do it again from scratch and you'll probably see what you did wrong.temaire said:
The Substitution Rule for Indefinite Integrals is a method of integration used to simplify integrals by substituting a new variable for the original variable. This allows for the integral to be rewritten in terms of the new variable, making it easier to solve.
To use the Substitution Rule, you must first identify the inner function and its derivative in the integral. Then, choose a new variable to substitute for the inner function, and replace all instances of the inner function with the new variable. Finally, rewrite the integral in terms of the new variable and solve.
The Substitution Rule is useful because it allows for the integration of more complicated functions by simplifying them into a form that is easier to solve. It also helps to avoid the use of complicated integration techniques such as integration by parts.
One common mistake is incorrect substitution, where the new variable is not chosen correctly or the original variable is not replaced correctly. Another mistake is forgetting to include the derivative of the inner function in the integral. It is also important to remember to change the limits of integration when using the Substitution Rule.
Yes, the Substitution Rule can also be used for definite integrals. However, the limits of integration must also be changed to match the new variable. This can be done by substituting the original limits into the new variable expression.