What Change Of Variables Is Suggested By An Integral Containing, When solving integrals, it is often helpful to change the variables in order to simplify the, General, what-change-of-variables-is-suggested-by-an-integral-containing, JPOSE
When solving integrals, it is often helpful to change the variables in order to simplify the integral and make it easier to solve. One common change of variable is suggested by an integral containing a square root.
Suppose we have an integral of the form:
∫ f(x)√(ax+b) dx
where a and b are constants. This integral can be difficult to solve directly, but we can make a change of variable to simplify it.
Let u = √(ax+b)
We can solve for x in terms of u:
u² = ax + b
x = (u² - b) / a
dx = 2u / a du
Substituting these expressions into the original integral, we get:
∫ f((u² - b) / a) * 2u / a du
This new integral may look more complicated, but it is actually easier to solve because it no longer contains the square root. We can now use integration techniques such as substitution or integration by parts to solve the integral.
Once we have found the solution in terms of u, we can substitute back in for x to get the final answer. This change of variable is particularly useful when the integrand contains a quadratic expression inside the square root, but it can also be used in other cases.
In summary, when faced with an integral containing a square root, we can often simplify it by making a change of variable to eliminate the square root. The suggested change of variable is to let u equal the expression inside the square root, solve for x in terms of u, substitute into the integral, and solve for the new integral in terms of u. This technique can make it easier to solve the integral and obtain the final answer.