Antiderivatives: Trying to Reverse
the Chain Rule (9/12/12)
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Any ideas about x 2(x 3 + 4)5 dx ??
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How about x ex^2 dx ?
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Try ln(x)/x dx
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But we’ve been lucky! Try sin(x 2) dx
What the 3 Examples Above
Had in Common:
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There was a chunk.
There was also a multiplier which was
the derivative of the chunk except for
possibly a missing constant multiplier.
This should allow us to see what an
antiderivative is, making the
appropriate adjustment for the missing
constant multiplier.
Different View of Same Idea:
Substitution Technique
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It’s called a “technique”, not a “rule”, because
it may or may not work (our text calls it a
“rule”, but I disagree!)
If there is a chunk, try calling the chunk u.
Compute du = (du/dx) dx
Replace all parts of the original expression
with things involving u (i.e., eliminate x).
If you were lucky/clever, the new expression
can be anti-differentiated easily.
An example of using “u –sub”
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What is x 2(x 3 + 4)5 dx ?
Set u = x 3 + 4 .
Then du/dx = 3x 2, so x 2 dx = (1/3)du.
Now replace equals with equals, totally
eliminating x ! We get (1/3) u 5 du .
Hence the answer is (1/3)(1/6)u 6 + C.
Replace u : Get (1/18)(x 3 + 4)6 + C.