Integral Of X Sin 2x Dx

Solving the Integral of x sin(2x) dx: A Comprehensive Guide

The integral ∫x sin(2x) dx is a classic example of integration by parts, a powerful technique used to solve integrals involving products of functions. This comprehensive guide will walk you through the process step-by-step, explaining the underlying principles and offering valuable insights into similar problems. We'll delve into the theory, demonstrate the solution, and explore related concepts to solidify your understanding.

Understanding Integration by Parts

The core principle behind integration by parts lies in the reverse product rule of differentiation. Recall that the derivative of a 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

This simplifies to:

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

Rearranging this equation, we arrive at the formula for integration by parts:

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

This formula allows us to solve integrals of products of functions by cleverly choosing u(x) and v'(x) such that the integral on the right-hand side is easier to solve than the original integral.

Applying Integration by Parts to ∫x sin(2x) dx

Now, let's apply this technique to solve our integral: ∫x sin(2x) dx.

First, we need to choose our u(x) and v'(x). A useful mnemonic to help with this choice is LIATE:

• Logarithmic functions
• Inverse trigonometric functions
• Algebraic functions (polynomials like x, x², etc.)
• Trigonometric functions (sin x, cos x, etc.)
• Exponential functions (eˣ, etc.)

The order in LIATE suggests a preference for choosing functions earlier in the list as u(x). In our case, x is an algebraic function, and sin(2x) is a trigonometric function. Following LIATE, we choose:

• u(x) = x
• v'(x) = sin(2x)

Now we need to find u'(x) and v(x):

• u'(x) = 1 (the derivative of x with respect to x)
• v(x) = -1/2 cos(2x) (the integral of sin(2x) with respect to x; remember the chain rule!)

Substitute these values into the integration by parts formula:

∫x sin(2x) dx = u(x)v(x) - ∫v(x)u'(x) dx

∫x sin(2x) dx = x(-1/2 cos(2x)) - ∫(-1/2 cos(2x))(1) dx

This simplifies to:

∫x sin(2x) dx = -1/2 x cos(2x) + 1/2 ∫cos(2x) dx

Now, we can easily integrate cos(2x):

∫cos(2x) dx = 1/2 sin(2x) + C (where C is the constant of integration)

Therefore, the final solution is:

∫x sin(2x) dx = -1/2 x cos(2x) + 1/4 sin(2x) + C

Verifying the Solution through Differentiation

A crucial step in solving any integral is verifying your solution. We can check our answer by differentiating it and seeing if we get back the original integrand, x sin(2x). Let's differentiate our solution:

d/dx [-1/2 x cos(2x) + 1/4 sin(2x) + C]

Using the product rule for the first term and the chain rule for the trigonometric terms:

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

= -1/2 cos(2x) + x sin(2x) + 1/2 cos(2x)

= x sin(2x)

As you can see, differentiation of our solution yields the original integrand, confirming the accuracy of our integration.

Extending the Understanding: Variations and Related Integrals

The techniques used to solve ∫x sin(2x) dx can be applied to a wide range of similar integrals. Let's explore some variations:

1. Integrals with Different Trigonometric Functions:

Consider integrals like ∫x cos(3x) dx or ∫x² sin(x) dx. The process remains the same: choose u(x) and v'(x) appropriately using LIATE, apply integration by parts, and simplify. For ∫x² sin(x) dx, you might need to apply integration by parts twice.

2. Integrals with Exponential Functions:

Integrals involving products of algebraic and exponential functions, such as ∫x eˣ dx, are also solved using integration by parts. Again, the LIATE rule will guide your choice of u(x) and v'(x).

3. Definite Integrals:

If you need to solve a definite integral, such as ∫[from a to b] x sin(2x) dx, simply evaluate the indefinite integral we found earlier at the limits of integration (b and a) and subtract the results.

Advanced Techniques and Considerations

For more complex integrals, more advanced techniques might be necessary, including:

• Tabular Integration: This method simplifies the process of repeated integration by parts, particularly useful when multiple iterations are required (e.g., ∫x³ eˣ dx).
• Trigonometric Substitution: If the integral involves trigonometric functions in a complex manner, trigonometric substitution might be the more efficient approach.
• Partial Fraction Decomposition: This technique is vital for integrals involving rational functions (ratios of polynomials).

Conclusion: Mastering Integration by Parts

The integral ∫x sin(2x) dx serves as an excellent foundational example for mastering integration by parts. By understanding the underlying principles, choosing u(x) and v'(x) strategically, and verifying your solution, you can confidently tackle a broad range of integration problems. Remember the LIATE rule as your guide and don't hesitate to practice with various examples to build your proficiency. With consistent practice and a clear understanding of the techniques discussed, you'll become adept at solving complex integrals and expanding your mathematical skills. The ability to solve integrals is crucial in various fields, from physics and engineering to economics and computer science. This foundational knowledge empowers you to tackle more complex problems and opens doors to a deeper understanding of mathematical concepts.