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x 1 Step 1 Recall the Inverse Substitution Rule where f and g are differentiable...

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Calculus

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x 1 Step 1 Recall the Inverse Substitution Rule where f and g are differentiable functions and g is one to one R x dx We are given the following dx 1 2 x x37 2 If f g t g t dt We note that x 1 2 x 1 Therefore the following entry from the Table of Trigonometric Substitutions is appropriate Expression It x a x a sec 8 0 8 or 0 3 2 2 1 2 a Substitution then a Identity sec 0 1 tan 0 Therefore we can let x sec 0 so dx de We also must make a substitution for the limits of integration in the definite integral Since x sec 8 we note that when x 2 0 Further when x 4 8 sec

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