Integration Of X Cos X

Integrating x cos x: A Comprehensive Guide

Integrating functions is a cornerstone of calculus, crucial for solving problems across numerous fields like physics, engineering, and economics. While some integrations are straightforward, others, like integrating x cos x, require specific techniques. This article provides a comprehensive guide to integrating x cos x, exploring different methods, explaining the underlying principles, and addressing frequently asked questions. Understanding this integral will not only enhance your calculus skills but also provide a solid foundation for tackling more complex integration problems.

Introduction: The Challenge of Integrating x cos x

The integral ∫x cos x dx presents a challenge because it's not a simple case of applying standard integration rules. We can't directly use the power rule or basic trigonometric integrals. Instead, we need a technique that elegantly handles the product of a polynomial function (x) and a trigonometric function (cos x). This technique is called integration by parts.

Understanding Integration by Parts

Integration by parts is a powerful technique derived from the product rule of differentiation. The product rule states that the derivative of the product of two functions, u(x) and v(x), is given by:

d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

Integrating both sides with respect to x, we get:

∫d/dx [u(x)v(x)] dx = ∫[u'(x)v(x) + u(x)v'(x)] dx

u(x)v(x) = ∫u'(x)v(x) dx + ∫u(x)v'(x) dx

Rearranging this equation, we obtain the integration by parts formula:

∫u(x)v'(x) dx = u(x)v(x) - ∫v(x)u'(x) dx

This formula allows us to transform a complex integral into a potentially simpler one. The key is to strategically choose u(x) and v'(x) to make the integral on the right-hand side easier to evaluate.

Integrating x cos x using Integration by Parts

Let's apply integration by parts to solve ∫x cos x dx.

1. Choose u(x) and v'(x): The choice of u(x) and v'(x) is crucial. A good rule of thumb is to choose u(x) as the function that simplifies when differentiated and v'(x) as the function that's easily integrable. In this case:

   • Let u(x) = x (because its derivative is simply 1)
   • Let v'(x) = cos x (because its integral is sin x)

2. Find u'(x) and v(x):

   • u'(x) = d/dx (x) = 1
   • v(x) = ∫cos x dx = sin x

3. Apply the Integration by Parts Formula:

   Substitute u(x), v(x), u'(x), and v'(x) into the integration by parts formula:

   ∫x cos x dx = x sin x - ∫sin x (1) dx

4. Evaluate the Remaining Integral:

   The integral ∫sin x dx is straightforward:

   ∫sin x dx = -cos x + C (where C is the constant of integration)

5. Combine the Results:

   Substitute this result back into the equation:

   ∫x cos x dx = x sin x - (-cos x) + C ∫x cos x dx = x sin x + cos x + C

Therefore, the integral of x cos x is x sin x + cos x + C.

A Deeper Look at the Choice of u(x) and v'(x)

The success of integration by parts often hinges on the judicious selection of u(x) and v'(x). While we chose u(x) = x and v'(x) = cos x above, let's explore what would happen if we made the opposite choice.

If we let u(x) = cos x and v'(x) = x, then:

• u'(x) = -sin x
• v(x) = x²/2

Applying the integration by parts formula:

∫x cos x dx = (cos x)(x²/2) - ∫(x²/2)(-sin x) dx

This leads to a more complicated integral, ∫(x²/2)(-sin x) dx, which is harder to solve than the original integral. This highlights the importance of strategic selection. The LIATE rule can be helpful:

• Logarithmic functions
• Inverse trigonometric functions
• Algebraic functions (polynomials)
• Trigonometric functions
• Exponential functions

This rule suggests choosing the function that comes first in the list as u(x). In our case, x (algebraic) comes before cos x (trigonometric), justifying our initial choice.

Generalizing the Method: Integrating xⁿ cos x

The method described above can be generalized to integrate functions of the form xⁿ cos x, where n is a positive integer. We can repeatedly apply integration by parts, reducing the power of x with each iteration. Let's look at the integral ∫x² cos x dx:

1. First Integration by Parts:

   • u(x) = x² => u'(x) = 2x
   • v'(x) = cos x => v(x) = sin x

   ∫x² cos x dx = x² sin x - ∫2x sin x dx

2. Second Integration by Parts:

   • u(x) = 2x => u'(x) = 2
   • v'(x) = sin x => v(x) = -cos x

   ∫2x sin x dx = 2x(-cos x) - ∫2(-cos x) dx = -2x cos x + ∫2 cos x dx

3. Final Integration:

   ∫2 cos x dx = 2 sin x + C

4. Combining the Results:

   ∫x² cos x dx = x² sin x - (-2x cos x + 2 sin x) + C ∫x² cos x dx = x² sin x + 2x cos x - 2 sin x + C

This illustrates how repeated application of integration by parts handles higher powers of x. Each iteration reduces the power of x by one, eventually leading to a solvable integral.

Exploring Definite Integrals

The techniques discussed also apply to definite integrals. For example, to evaluate ∫(from 0 to π) x cos x dx, we use the indefinite integral we derived earlier:

∫(from 0 to π) x cos x dx =

Evaluating at the limits of integration:

[π sin π + cos π] - [0 sin 0 + cos 0] = [0 + (-1)] - [0 + 1] = -2

Thus, the definite integral from 0 to π is -2.

Frequently Asked Questions (FAQ)

Q1: Are there other methods to integrate x cos x?

A1: While integration by parts is the most straightforward and commonly used method, other techniques might be applicable depending on the context. However, for this specific integral, integration by parts is the most efficient approach.

Q2: What if the integral involves x sin x instead of x cos x?

A2: The approach is very similar. You would still use integration by parts, but the roles of sin x and cos x would be reversed. The resulting integral would be different but would still be solvable using this technique.

Q3: Can this method be extended to integrals involving other trigonometric functions?

A3: Yes, integration by parts is a general technique applicable to integrals involving various combinations of functions, including other trigonometric functions like sin x, tan x, sec x, etc., and their products with polynomial or exponential functions. The choice of u(x) and v'(x) will dictate the success and efficiency of the method.

Q4: What if 'x' is replaced by a more complex algebraic expression?

A4: If 'x' is replaced by a more complex algebraic expression, the integration by parts process remains fundamentally the same. However, calculating u'(x) and integrating v'(x) might become more involved. This might require using other integration techniques in conjunction with integration by parts.

Conclusion

Integrating x cos x, seemingly a daunting task at first, becomes manageable and even elegant through the application of integration by parts. Understanding this technique not only solves this specific integral but provides a foundation for tackling a broader range of integration problems involving products of functions. By mastering the strategic selection of u(x) and v'(x) and practicing iterative applications of the method, you'll significantly enhance your calculus skills and confidence in solving complex integration problems. Remember the importance of careful steps, proper substitution, and diligent attention to detail. Practice is key to mastering this valuable tool in your mathematical arsenal.