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To evaluate the integral of the function tan(x) with respect to x, we can employ a substitution method by letting u equal cos(x). This substitution is particularly useful because it simplifies the integral by transforming the trigonometric function into a more manageable form. By differentiating u with respect to x, we find that du = -sin(x)dx, which allows us to express dx in terms of du. Consequently, we can rewrite the integral in terms of u, facilitating the integration process.
Next, we recognize that tan(x) can be expressed as sin(x)/cos(x). Substituting u for cos(x) leads us to rewrite sin(x) in terms of u as well. Since sin(x) can be expressed as √(1 – u²) when considering the Pythagorean identity, we can substitute this into our integral. Thus, the integral of tan(x)dx transforms into an expression involving u, which can be integrated more easily. This step is crucial as it allows us to focus on a single variable, simplifying the overall computation.
Finally, after performing the integration with respect to u, we will need to revert back to the original variable x by substituting u back in terms of cos(x). This final step ensures that our solution is expressed in the context of the original integral. The result will yield the antiderivative of tan(x) in terms of x, completing the integration process. This method not only demonstrates the utility of substitution in calculus but also highlights the interconnectedness of trigonometric functions and their derivatives.
