Inside Our Calculus Courseware: Trigonometric Substitutions

The word “Trigonometric” by itself scares students. Combining it with “Substitutions and Evaluation” is downright terrifying. After all, the student must select the appropriate substitution, transform the integrand from an algebraic to a trigonometric expression, make the appropriate change in limits of integration (or rewrite their antiderivative in terms of the original variable), and finally evaluate the antiderivative. There are pitfalls everywhere along the way. One thing students often fail to do is carry out the last step and evaluate the integral because they’re so relieved to have found the antiderivative.

Sample Problem #1

Below is an example of this problem type and ways we show students how to avoid those common pitfalls:



In the Practice mode, students have access to learning aids to help them understand how to tackle each problem. For example, they can choose Step-by-Step in the Tutor area.

This tool provides a step-by-step breakdown of the problem, walking the student through the problem in manageable pieces. While it provides plenty of guidance, the Step-by-Step portion does ask the student to input the results of each step so they are learning as they go.

In Step 1, since the integrand does not exactly match any of the expressions corresponding to a trigonometric substitution, specifically the expression under the radical, the student is asked to identify the equivalent form of that expression after it has been rewritten by completing the square.


In Step 2, the student will identify the limits of integration after the first change in variable.


In Step 3, the student will identify the trigonometric substitution.


In Step 4, the student calculates the differential dt in terms of the new variable θ after the substitution in Step 3.

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In Step 5, the student will identify the limits of integration in terms of θ resulting from the trigonometric substitution.


In Step 6, the student is prompted to simplify the integrand if the absolute value can be removed. The condition for which this is possible is verified.


In Step 7, the student will find and evaluate the antiderivative. There is no need to rewrite the antiderivative in terms of the original variable since the limits of integration have been rewritten at each step in terms of the new variables when new variables were introduced. Because of this, taking the antiderivative and evaluating it is straightforward.



Sample Problem #2

Students often are so relieved at finally having found the antiderivative, they fail to take the final step and evaluate that antiderivative for a definite integral. The following Explain Error example notes when this occurs and prompts the student to take that final step.

The correct but unevaluated antiderivative is entered.


Students can select the Explain Error option to receive precise feedback from the system’s artificial intelligence. This tool anticipates and diagnoses specific errors, stopping students in their tracks and showing them not only that their answer is incorrect, but why it is incorrect.

Here, we note the correct but unevaluated antiderivative has been entered as the answer.


The student then returns to Practice mode, evaluates the result at the limits of integration, and completes the question.



Calc Book and Computer


Interested in seeing more of the calculus question bank? Contact us today at or 1-800-426-9538 to get free access to the student courseware!

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