Calculating The Indefinite Integral Of Sec²(x)tan²(x)

#SEO Title: Integrating sec²(x)tan²(x): A Step-by-Step Guide

In the realm of calculus, indefinite integrals often present intriguing challenges that require a blend of trigonometric identities, substitution techniques, and a solid understanding of fundamental integration rules. This article delves into the process of calculating the indefinite integral of sec²(x)tan²(x), providing a comprehensive step-by-step guide to unraveling this mathematical expression. We will explore the underlying principles, employ strategic substitutions, and ultimately arrive at the solution, ensuring we include the crucial '+C' to represent the constant of integration.

Understanding the Integral: ∫ sec²(x)tan²(x) dx

The indefinite integral ∫ sec²(x)tan²(x) dx beckons us to find a function whose derivative is sec²(x)tan²(x). To tackle this, we need to recognize the interplay between trigonometric functions and their derivatives. The key here lies in identifying a suitable substitution that simplifies the integral. Remember, the goal of integration is to reverse the process of differentiation, so we're essentially looking for a function that, when differentiated, yields our integrand. Let's first break down the components of the integrand. We have sec²(x), which is the square of the secant function, and tan²(x), which is the square of the tangent function. The derivative of the tangent function is sec²(x), which is a crucial piece of information that will guide our substitution strategy. This relationship suggests that substituting u = tan(x) might be a fruitful approach. By doing so, we transform the integral into a simpler form that we can readily integrate using the power rule. The power rule states that the integral of x^n is (x^(n+1))/(n+1) + C, where C is the constant of integration. This is a fundamental rule in calculus and will be instrumental in solving our integral once we've made the appropriate substitution. Now, let's proceed with the substitution and see how it unfolds.

The Substitution Method: A Powerful Tool

To effectively evaluate the indefinite integral ∫ sec²(x)tan²(x) dx, the substitution method emerges as a powerful technique. This method allows us to simplify complex integrals by introducing a new variable that transforms the integrand into a more manageable form. In this specific case, observing the relationship between the tangent function and its derivative, the secant squared function, leads us to a strategic substitution. Let's set u = tan(x). This substitution is motivated by the fact that the derivative of tan(x) is sec²(x), which is also present in our integrand. This connection is key to simplifying the integral. Now, we need to find the differential du. Differentiating both sides of u = tan(x) with respect to x, we get du/dx = sec²(x). Multiplying both sides by dx, we obtain du = sec²(x) dx. This is a crucial step, as it allows us to replace sec²(x) dx in the original integral with the much simpler du. With this substitution in hand, we can rewrite the integral in terms of u. Replacing tan(x) with u and sec²(x) dx with du, the integral ∫ sec²(x)tan²(x) dx transforms into ∫ u² du. This new integral is significantly simpler than the original and can be easily solved using the power rule of integration. The power rule states that the integral of x^n is (x^(n+1))/(n+1) + C, where C is the constant of integration. Applying this rule to our integral in terms of u, we can find the antiderivative. Let's proceed with applying the power rule to ∫ u² du.

Applying the Power Rule: Integrating u²

With the integral transformed into ∫ u² du, we can now leverage the power rule of integration to find its antiderivative. The power rule, a cornerstone of integral calculus, states that for any real number n ≠ -1, the integral of xⁿ dx is (x^(n+1))/(n+1) + C, where C represents the constant of integration. In our case, we have u² which corresponds to xⁿ with n = 2. Applying the power rule directly, we increase the exponent by 1, resulting in u³, and then divide by the new exponent, 3. This yields (u³)/3 as the antiderivative of u². Therefore, the integral of u² du is (u³)/3 + C. This is a straightforward application of the power rule, and it brings us closer to the final solution of our original integral. However, we're not quite done yet. Remember, our original integral was in terms of x, and we made a substitution to simplify it. Now that we've found the antiderivative in terms of u, we need to reverse the substitution to express the result in terms of x. This is a crucial step in the substitution method, as it ensures that our final answer is in the same variable as the original problem. To do this, we simply replace u with its original expression in terms of x, which was u = tan(x). Let's perform this back-substitution and see what our result looks like.

Reversing the Substitution: Back to x

Having integrated ∫ u² du to obtain (u³)/3 + C, the next crucial step is reversing the substitution to express the result in terms of the original variable, x. Recall that we initially made the substitution u = tan(x). Now, we simply replace u in our antiderivative with tan(x). This gives us (tan³(x))/3 + C. This expression represents the indefinite integral of sec²(x)tan²(x) with respect to x. The constant of integration, C, is a vital component of the answer, as it acknowledges the fact that the derivative of a constant is zero. This means there are infinitely many functions whose derivative is sec²(x)tan²(x), differing only by a constant term. Therefore, including '+C' is essential for a complete and accurate solution to the indefinite integral. Now, let's present the final answer, clearly stating the indefinite integral and the constant of integration.

The Final Solution: The Indefinite Integral

After meticulously applying the substitution method and the power rule, we arrive at the final solution for the indefinite integral of sec²(x)tan²(x). We have successfully demonstrated that: ∫ sec²(x)tan²(x) dx = (tan³(x))/3 + C This elegant expression represents the family of functions whose derivative is sec²(x)tan²(x). The term (tan³(x))/3 is the antiderivative we found through our integration process, and '+C' signifies the constant of integration. This constant is a crucial reminder that there are infinitely many solutions to an indefinite integral, each differing by a constant value. In conclusion, we have navigated the intricacies of this integral, employing a strategic substitution and the power rule to arrive at the solution. This process highlights the power and elegance of calculus techniques in solving complex mathematical problems.