When would you use u substitution twice?

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When we are reversing a contrariantiation that had the compound of three functions. Here is one issue.

##int sin^4(7x)cos(7x)dx##

Let ##u=7x##. This makes ##du = 7dx## and our well can be rewritten:

##1/7 int sin^4ucosudu = 1/7int(sinu)^4cosudu##

To quit using ##u## to medium two contrariant things in one discourse, we'll use another shifting (##t, v, w## are all approved choices)

Let ##w=sinu##, so we entertain ##dw = cosudu## and our well becomes:

##1/7intw^4dw##

We the consolidate and back-substitute:

##1/7intw^4dw = 1/35 w^5 +C##

## = 1/35 sin^5u +C##

## = 1/35 sin^5 7x +C##

If we cohibit the rejoinder by contrariantiating, we'll use the twice.

##d/dx((sin(7x))^5) = 5(sin(7x))^4*d/dx(sin(7x))##

## = 5(sin(7x))^4*cos(7x)d/dx(7x)##

## = 5(sin(7x))^4*cos(7x)*7##

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