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Chapter 5: Q 57. (page 429)
Solve the integral:
b
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Why don鈥檛 we need to have a square root involved in order to apply trigonometric substitution with the tangent? In other words, why can we use the substitution when we see , even though we can鈥檛 use the substitution unless the integrand involves the square root of? (Hint: Think about domains.)
Show by differentiating (and then using algebra) that and are both antiderivatives of . How can these two very different-looking functions be an antiderivative of the same function?
Solvethe following two ways:
(a) with the substitution
(b) with the trigonometric substitution x = 2 tan u.
Solve the integral:.
Suppose you use polynomial long division to divide p(x) by q(x), and after doing your calculations you end up with the polynomial as the quotient above the top line, and the polynomial 3x 鈭� 1 at the bottom as the remainder. Then
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