MAT136H1 Lecture : 5.5 Integration & Anti-derivatives Method of Substitution Question #4 (Medium)
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Usually substitutable functions are hidden inside distinctive group of functions, such as trigonometric, exponential powers, square roots, rational functions in the denominator, etc. When using the method of substitution, make sure to not only apply the most appropriate substitution, but also to change the original interval to the new interval that match the substitution. When the change of the interval is made from the start, substituting back in the original variable is not necessary. This is the preferred and easier approach to solving definite integrals. Evaluate the definite integral using the method of substitution. Notice the functions in the numerator and the denominator. The key is to link these two after the form of chain rule, so that the most appropriate substitution is made. As for the interval, when , then , and when , then . Thus, substituting the variables and , as well as changed interval into the definite integral:
