121616949-math.172 - given function so that it is written...

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158 Chapter 8 Techniques of Integration Recall that one benefit of the Leibniz notation is that it turns out that what looks like ordinary arithmetic gives the correct answer, even if something more complicated is going on. For example, in Leibniz notation the chain rule is dy dx = dy dt dt dx . The same is true of our current expression: Z x 2 - 2 u du dx dx = Z x 2 - 2 u du. Now we’re almost there: since u = 1 - x 2 , x 2 = 1 - u and the integral is Z - 1 2 (1 - u ) u du. It’s no coincidence that this is exactly the integral we computed in ( 8 . 1 ), we have simply renamed the variable u to make the calculations less confusing. Just as before: Z - 1 2 (1 - u ) u du = 1 5 u - 1 3 u 3 / 2 + C. Then since u = 1 - x 2 : Z x 3 p 1 - x 2 dx = 1 5 (1 - x 2 ) - 1 3 (1 - x 2 ) 3 / 2 + C. To summarize: if we suspect that a given function is the derivative of another via the chain rule, we let u denote a likely candidate for the inner function, then translate the
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Unformatted text preview: given function so that it is written entirely in terms of u , with no x remaining in the expression. If we can integrate this new function of u , then the antiderivative of the original function is obtained by replacing u by the equivalent expression in x . Even in simple cases you may prefer to use this mechanical procedure, since it often helps to avoid silly mistakes. For example, consider again this simple problem: Z 2 x cos( x 2 ) dx. Let u = x 2 , then du/dx = 2 x or du = 2 x dx . Since we have exactly 2 x dx in the original integral, we can replace it by du : Z 2 x cos( x 2 ) dx = Z cos u du = sin u + C = sin( x 2 ) + C....
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