When would you use u substitution twice?

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Answer 1 · 2015-07-01T13:05:13

When we are reversing a differentiation that had the composition of three functions. Here is one example.

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

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

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

To avoid using #u# to mean two different things in one discussion, we'll use another variable (#t, v, w# are all popular choices)

Let #w=sinu#, so we have #dw = cosudu# and our integral becomes:

#1/7intw^4dw#

We the integrate and back-substitute:

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

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

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

If we check the answer by differentiating, we'll use the chain rule 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#