Exploring the Integral of x sin x: A Comprehensive Guide

Integration by parts is a powerful technique in calculus, often used to integrate the product of two functions. The integral of x sin x can be solved using this method. In this article, we walk you through the step-by-step process of solving this integral, as well as provide a deeper understanding of the integration by parts formula.

Understanding Integration by Parts

Integration by parts is a theorem that allows the integration of a product of functions. The formula for integration by parts is:

∫udv uv - ∫vdu

Where u and dv are chosen from the original integral based on the technique of "reverse chain rule."

Solving the Integral of x sin x

To find the integral of x sin x, we use integration by parts. Let's begin by selecting u and dv as follows:

Set u x. Then, the derivative of u is du dx. Set dv sin x dx. Then, the integral of dv is v -cos x.

Applying the integration by parts formula:

∫x sin x dx uv - ∫v du

Substitute the chosen values:

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

This simplifies to:

∫x sin x dx x cos x ∫ cos x dx

Now, we need to integrate cos x:

∫ cos x dx sin x

Therefore, our equation becomes:

∫x sin x dx x cos x sin x C

Where C is the constant of integration.

Verification and Further Insights

To verify the solution, we can differentiate the result to ensure we get back the original integrand:

d/dx (x cos x sin x C) -x sin x cos x cos x x sin x

The differentiation confirms the correctness of our solution.

Additional Examples and Practical Applications

Integration by parts is applicable in many practical scenarios. For instance, it can be used in physics for calculating work done by a force, in engineering for determining electric fields, and in economics for integrating cost functions.

Here is another example:

∫ x e^x dx

Choose u x and dv e^x dx. Then, du dx and v e^x.

∫ x e^x dx x e^x - ∫ e^x dx

x e^x - e^x C

This example further illustrates how to apply integration by parts in similar problems.

Conclusion

Integration by parts is an essential technique in calculus. By understanding and practicing the steps involved, you can solve a wide range of integration problems. The integral of x sin x is a classic example that highlights the power and utility of this method.