Evaluating the integral of sin 2x cos 2x involves recognizing a fundamental trigonometric relationship that simplifies the problem significantly. The expression within the integral represents a product of sine and cosine with identical arguments, which immediately suggests the use of a double-angle identity. By applying the principle that sin(2θ) = 2 sin θ cos θ, we can adapt this to the specific case where the angle is 2x, leading to a direct simplification. This initial step transforms the integral into a more manageable form, allowing for straightforward integration techniques to be applied. The process highlights the importance of algebraic manipulation in calculus to reduce complex expressions into basic forms.
Understanding the Trigonometric Identity
The core of solving this integral lies in the double-angle identity for sine, which states that sin(4x) = 2 sin(2x) cos(2x). By isolating the product sin(2x) cos(2x), we can express it as a fraction of sin(4x), specifically (1/2) sin(4x). This transformation is the critical link between the original problem and a simpler integral. Instead of dealing with the product of two different trigonometric functions, we work with a single sine function with a modified coefficient and argument. This adjustment is not merely a mathematical trick but a logical application of established identities that streamlines the entire integration process.
Method 1: Using the Double-Angle Formula
To solve the integral using the first method, we substitute the identity directly into the integral expression. The integral of sin 2x cos 2x dx becomes the integral of (1/2) sin(4x) dx. The constant coefficient 1/2 can be moved outside the integral, leaving us with (1/2) times the integral of sin(4x) dx. The integral of sin(u) with respect to u is -cos(u) + C, and applying this rule requires adjusting for the coefficient of x. By integrating, we find that the solution is -1/8 cos(4x) + C. This result is derived purely through algebraic substitution and the standard integral of the sine function.
Method 2: Using u-Substitution
An alternative approach to solving the integral of sin 2x cos 2x dx is to use a u-substitution without initially applying the double-angle identity. In this method, we let u equal sin(2x), which means the derivative du/dx is 2 cos(2x). We can then rearrange this to express cos(2x) dx in terms of du, specifically (1/2) du. Substituting these terms into the original integral converts it into the integral of u times (1/2) du. This simplifies to (1/2) times the integral of u du, which is a basic power rule integral. Solving this yields (1/2) * (u^2 / 2) + C, or u^2 / 4 + C. Finally, substituting sin(2x) back in for u gives the solution (1/4) sin^2(2x) + C.
It is important to note that while the expressions -1/8 cos(4x) + C and (1/4) sin^2(2x) + C appear different, they are equivalent results of the integration process. This equivalence can be proven using the Pythagorean identity and the double-angle formula for cosine, where cos(4x) = 1 - 2 sin^2(2x). Substituting this into the first solution shows that it simplifies to the second solution, differing only by a constant value. This consistency validates both integration methods and demonstrates that multiple paths can lead to the correct antiderivative in calculus.
Verification and Graphical Interpretation
More perspective on Integral of sin 2 x cos 2 x can make the topic easier to follow by connecting earlier points with a few simple takeaways.
