Final answer:
The substitution method is a technique used in integration to simplify the problem by substituting a new variable in place of the original variable. Here are the steps to use the substitution method: Identify a suitable substitution, Let the new variable be equal to the function found, Replace the original variable, Carry out the integration, Finally, convert the answer back
Step-by-step explanation:
The substitution method is a technique used in integration to simplify the problem by substituting a new variable in place of the original variable. Here are the steps to use the substitution method:
Identify a suitable substitution by looking for a function and its derivative in the integrand.
Let the new variable be equal to the function found in step 1.
Replace the original variable and its derivative with the new variable and its derivative in the integrand.
Carry out the integration using the new variable.
Finally, convert the answer back to the original variable if necessary.
For example, suppose we have the integral ∫(2x-1)√(x³-x) dx. By letting u = x³-x, we can simplify the integrand to ∫(2u-1)√u du.
After finding the antiderivative with respect to u, we can convert the answer back to x by substituting u with x³-x.