Mastering the integral of sin 2x cosx requires a blend of trigonometric insight and strategic substitution. This specific form appears frequently in calculus courses and engineering applications, where simplifying the integrand is the first critical step. By recognizing the double-angle identity, the expression transforms into a more manageable polynomial in terms of sine.
Understanding the Core Trigonometric Identity
The foundation of solving this integral lies in the double-angle formula for sine, where sin 2x is defined as 2 sin x cos x. Substituting this identity into the original integral converts the problem from a product of three functions into a product of sine and cosine terms that share a direct relationship. This conversion is essential because it sets up the structure required for the u-substitution method to work effectively, turning a complex trigonometric integral into a simple power rule problem.
Step-by-Step Integration Process
To solve the integral, we follow a clear algebraic path. First, rewrite the integral using the identity: ∫ (2 sin x cos x) cos x dx. Next, combine the cosine terms to get 2 ∫ sin x cos² x dx. At this stage, we observe that the derivative of cos x is -sin x, which is present in the integrand, making u-substitution the optimal strategy. Let u = cos x, which implies du = -sin x dx.
Substituting these terms changes the integral to -2 ∫ u² du. Applying the power rule for integration, we integrate u² to get u³/3, resulting in -2 * (u³/3) + C. Finally, we substitute back cos x for u to return to the original variable, yielding the final solution: -2/3 cos³ x + C. This method ensures accuracy and provides a systematic approach to similar trigonometric products.
Verification Through Differentiation
To confirm the validity of the result, we can differentiate the solution to see if we return to the original integrand. Taking the derivative of -2/3 cos³ x + C involves the chain rule. The derivative of cos³ x is 3 cos² x * (-sin x), which simplifies to -3 cos² x sin x. Multiplying this by the constant -2/3 yields 2 cos² x sin x.
Using the double-angle identity in reverse, 2 sin x cos x equals sin 2x, so the expression becomes sin 2x cos x. Since this matches the original function under the integral sign, the integration is verified as correct. This step is crucial for building confidence in the solution and understanding the relationship between differentiation and integration.
Alternative Methods and Strategic Insights
While the identity method is standard, advanced students might explore alternative approaches for verification. One could expand sin 2x using exponential forms or employ integration by parts; however, these methods are generally more complex. The most efficient path always involves reducing the problem to a polynomial through substitution. Recognizing the derivative of the inner function (cos x) within the integrand is the key to selecting the correct substitution.
Understanding when to apply trigonometric identities versus when to use direct substitution is a critical skill in integral calculus. This problem serves as an excellent example of how a single identity can unlock the solution, transforming a potentially confusing product of functions into a straightforward power rule integration. Practicing this sequence reinforces the logical flow required in higher-level mathematics.
