How do you find the du in a substitution rule for integrals?

substitution-rule
integrals

#1

how do you find the du in a substitution rule for integrals?

Answer:

With substitution, you have to pick what u is. Its the reverse operation of the chain rule in differential calculus. With an integral like

2x(x^2 + 1)^(3/2)

if we pick u = x^2 + 1, we then have to determine what du is. Taking the derivative of u gives us du = 2x dx

then du/dx = 2x, so our integral becomes u^(3/2)

Then we take the integral and substitute u back in

= 2/5 u ^(5/2) + c
= 2/5(x^2 + 1)^(5/2) + c

With the chain rule we have to take the derivative of the outside function times the derivative of whats inside. When you are using substitution, pick u in a way that u’s derivative is seen in the integral differing at most by a constant