Integral Of Sin 2x X 2

The integral ofsin(2x) multiplied by x², written as ∫ sin(2x) * x² dx, represents a fundamental problem in calculus involving the combination of trigonometric functions and polynomial terms. This specific integral is a classic example where the technique of integration by parts proves essential. It requires careful application of the formula ∫u dv = u v - ∫v du, strategically selecting parts of the product to simplify the resulting integral step-by-step. Mastering this process builds a crucial foundation for tackling more complex integrals involving products of polynomials and trigonometric or exponential functions, highlighting the interconnectedness of different mathematical concepts.

Introduction: Understanding the Integral of sin(2x) * x² The integral ∫ sin(2x) * x² dx seeks the antiderivative of the product of the sine function with a double argument (2x) and a quadratic polynomial term (x²). This type of integral is a staple in calculus courses, often appearing in problems involving areas, volumes, or solutions to differential equations. The presence of the polynomial term x² suggests that integration by parts, a method derived from the product rule for differentiation, is the most appropriate strategy. The double angle in the sine function introduces a constant factor that must be handled carefully during the integration process. Successfully evaluating this integral requires a systematic approach, breaking down the problem into manageable steps while maintaining attention to the algebraic and trigonometric manipulations involved. The final result provides a function whose derivative, when differentiated, returns the original integrand sin(2x) * x², confirming the correctness of the solution.

Steps to Solve ∫ sin(2x) * x² dx

1. Identify Integration by Parts: Recognize that the product of sin(2x) and x² requires integration by parts. Set u and dv accordingly.

   • Choice 1: Let u = x² and dv = sin(2x) dx. This is often a good choice because differentiating the polynomial simplifies it.
   • Choice 2: Let u = sin(2x) and dv = x² dx. Differentiating the trig function might lead to more complex integrals involving cos(2x) and polynomials.
   • Recommendation: Choice 1 is generally preferred for this integrand as it simplifies the polynomial part.

2. Compute du and v:

   • If u = x², then du = 2x dx.
   • If dv = sin(2x) dx, then v = ∫ sin(2x) dx = (-1/2) cos(2x). (Recall: The derivative of cos(2x) is -2sin(2x), so the integral of sin(2x) is (-1/2) cos(2x).)

3. Apply Integration by Parts Formula:

   • ∫ u dv = u v - ∫ v du
   • Substituting: ∫ sin(2x) * x² dx = (x²) * [(-1/2) cos(2x)] - ∫ [(-1/2) cos(2x)] * (2x dx)

4. Simplify the Expression:

   • = (-1/2) x² cos(2x) - ∫ [(-1/2) cos(2x)] * 2x dx
   • = (-1/2) x² cos(2x) - ∫ [-x cos(2x)] dx
   • = (-1/2) x² cos(2x) + ∫ x cos(2x) dx

5. Solve the New Integral (∫ x cos(2x) dx): This new integral also requires integration by parts.

   • Set u = x and dv = cos(2x) dx.
   • du = dx
   • v = ∫ cos(2x) dx = (1/2) sin(2x) (Recall: The derivative of sin(2x) is 2cos(2x), so the integral of cos(2x) is (1/2) sin(2x).)

6. Apply Integration by Parts Again:

   • ∫ x cos(2x) dx = u v - ∫ v du
   • = (x) * [(1/2) sin(2x)] - ∫ [(1/2) sin(2x)] * dx
   • = (1/2) x sin(2x) - (1/2) ∫ sin(2x) dx

7. Integrate sin(2x) Again:

   • ∫ sin(2x) dx = (-1/2) cos(2x) (As established in Step 2)

8. Combine All Results:

   • ∫ x cos(2x

Continuing from the point where thesecond integration by parts was applied:

8. Complete the Second Integration by Parts: Substituting the results from step 7 into the expression from step 6:

   • ∫ x cos(2x) dx = (1/2) x sin(2x) - (1/2) ∫ sin(2x) dx
   • = (1/2) x sin(2x) - (1/2) * [ (-1/2) cos(2x) ] + C₁
   • = (1/2) x sin(2x) + (1/4) cos(2x) + C₁

9. Substitute Back into the First Integration by Parts Result: Now substitute this result back into the expression obtained in step 4:

   • ∫ sin(2x) * x² dx = (-1/2) x² cos(2x) + ∫ x cos(2x) dx
   • = (-1/2) x² cos(2x) + [ (1/2) x sin(2x) + (1/4) cos(2x) ] + C
   • = (-1/2) x² cos(2x) + (1/2) x sin(2x) + (1/4) cos(2x) + C

10. Combine Like Terms: Group the terms containing cos(2x):

    • = cos(2x) [ - (1/2) x² + 1/4 ] + (1/2) x sin(2x) + C
    • = cos(2x) [ - (1/2) x² + 1/4 ] + (1/2) x sin(2x) + C

11. Write the Final Answer: The indefinite integral is: ∫ sin(2x) * x² dx = (-1/2) x² cos(2x) + (1/2) x sin(2x) + (1/4) cos(

(2x) + C

Conclusion

The integral ∫ x² sin(2x) dx is evaluated successfully through a two-step application of integration by parts. The initial choice of u = x² (and dv = sin(2x) dx) strategically reduces the polynomial's degree, converting the problem into the simpler integral ∫ x cos(2x) dx. A second application then resolves this remaining integral. The final result combines polynomial and trigonometric functions, demonstrating the systematic power of integration by parts for products of polynomials and trigonometric functions. The constant of integration, C, accounts for the indefinite nature of the original integral.

Building upon these insights, the interplay of algebraic manipulation and analytical rigor continues to shape mathematical understanding, offering solutions that transcend mere calculation. Such processes underscore the enduring relevance of foundational techniques in addressing complex challenges. Thus, mastery remains a cornerstone for advancing knowledge across disciplines.