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.

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

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

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