notes Integration by substitution

Thomas' Calculus: Early Transcendentals

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Integration by substitution The best general advice is this: substitute for the troublesome part of the integral but don’t be too greedy. For example, in Z e t dt t , the e t is troublesome, but you should substitute just for the t ; let y = t . There is a natural tendency immediately to differentiate after a substitution is made, but this is not always wise. In the example above, we can get away with this: d t dt = 1 2 t and we were lucky, since 1 t occurs in the integral. But a safer procedure is to express the old variable in terms of the new and then differentiate. In our example, y = t so t = y 2 and dt = 2 y dy . Similarly, in Z dx x - x the troublesome part is x . Let y = x , so x = y 2 and dx = 2 y dy . Trigonometric functions and cofuntions The following pairs of trigonometric functions are called co functions of each other, because each function in a pair is the other function of the co mplementary angle: sin
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Unformatted text preview: θ = cos( π/ 2-θ ) cos θ = sin( π/ 2-θ ) tan θ = cot( π/ 2-θ ) cot θ = tan( π/ 2-θ ) sec θ = csc( π/ 2-θ ) csc θ = sec( π/ 2-θ ) Notice that d ( π/ 2-θ ) =-dθ . This has the following consequence: In any integration or differentiation formula involving trigonometric functions of θ alone, we can replace all trigonometric functions by their cofunctions and change the overall sign. The use of this rule cut memorization in half. For example, Z cot θ dθ = Z cos θ dθ sin θ = ln | sin θ | + C , so Z tan θ dθ =-ln | cos θ | + C . If we remember that Z sec θ dθ = ln | sec θ + tan θ | + C , then we automatically have Z csc θ dθ =-ln | csc θ + cot θ | + C . And if we remember that d sec θ = sec θ tan θ dθ , then we know that d csc θ =-csc θ cot θ dθ ....
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This note was uploaded on 02/12/2008 for the course MATH 104 taught by Professor Nelson during the Fall '07 term at Princeton.

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