In other words, when solving integration problems, we make appropriate substitutions to obtain an integral that becomes much simpler than the original integral. Thus the integral becomes ∫ 0 5 1 2 u 5 d u ∫ 0 5 1 2 u 5 d u and this integral is much simpler to evaluate. Then d u = 2 x d x d u = 2 x d x or x d x = 1 2 d u x d x = 1 2 d u and the limits change to u = g ( 2 ) = 2 2 − 4 = 0 u = g ( 2 ) = 2 2 − 4 = 0 and u = g ( 3 ) = 9 − 4 = 5. When evaluating an integral such as ∫ 2 3 x ( x 2 − 4 ) 5 d x, ∫ 2 3 x ( x 2 − 4 ) 5 d x, we substitute u = g ( x ) = x 2 − 4. Recall from Substitution Rule the method of integration by substitution. 5.7.4 Evaluate a triple integral using a change of variables.5.7.3 Evaluate a double integral using a change of variables.5.7.2 Compute the Jacobian of a given transformation.5.7.1 Determine the image of a region under a given transformation of variables.
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