Integral Of X Cos X 2

Solving the Integral of x cos(x²)

This article will guide you through the process of solving the integral ∫x cos(x²) dx. This integral requires a technique known as u-substitution, a fundamental method in calculus for simplifying integrals. Understanding this process is crucial for mastering integration techniques and tackling more complex problems. This guide provides a step-by-step solution, making it easy to understand even for beginners.

Understanding the Problem and Choosing a Strategy

The integral ∫x cos(x²) dx presents a challenge because we don't have a readily available antiderivative for the expression x cos(x²). Direct integration is not feasible. However, the presence of x (the derivative of x²) alongside the trigonometric function suggests that u-substitution is the appropriate approach. This method involves substituting a portion of the integrand with a new variable, 'u', simplifying the integral, and then substituting back to express the result in terms of the original variable, 'x'.

Step-by-Step Solution using u-Substitution

1. Choose the substitution: Let u = x². This choice is strategic because the derivative of u, which is du = 2x dx, appears in the integrand (or a multiple of it).

2. Express dx in terms of du: From du = 2x dx, we can solve for dx: dx = du / (2x).

3. Substitute: Substitute u for x² and du / (2x) for dx in the original integral:

   ∫x cos(x²) dx = ∫x cos(u) * (du / (2x))

4. Simplify: Notice that the 'x' in the numerator cancels with the 'x' in the denominator:

   ∫x cos(u) * (du / (2x)) = (1/2) ∫cos(u) du

5. Integrate: The integral of cos(u) is simply sin(u):

   (1/2) ∫cos(u) du = (1/2) sin(u) + C

6. Back-substitute: Substitute x² back in for u to express the result in terms of x:

   (1/2) sin(u) + C = (1/2) sin(x²) + C

Therefore, the solution to the integral ∫x cos(x²) dx is (1/2)sin(x²) + C, where C is the constant of integration.

Key Concepts Revisited

• U-Substitution: This technique involves substituting a part of the integrand with a new variable to simplify the integration process. The successful application of u-substitution often hinges on identifying a suitable substitution and its derivative within the original integral.

• Constant of Integration (C): It's crucial to remember the constant of integration (C) when solving indefinite integrals. This constant accounts for the fact that the derivative of a constant is always zero. Different constants lead to different antiderivatives for the same function.

• Chain Rule (in reverse): U-substitution can be viewed as the reverse application of the chain rule of differentiation.

Practical Applications and Further Exploration

Understanding the integral of x cos(x²) is foundational for various applications in physics, engineering, and other scientific fields where integration is essential, particularly in dealing with oscillatory systems or wave phenomena. Furthermore, mastering u-substitution opens doors to solving far more complex integrals. Practice with various examples will solidify your understanding and build your problem-solving skills.