Hey guys! Stuck on how to solve the integral of xsin(x) dx? Don't worry, you're not alone! This is a classic problem in calculus that often trips people up, but with the right approach, it becomes super manageable. We're going to break down the solution step-by-step, making sure you understand not just the what, but also the why behind each move. So, grab your favorite beverage, settle in, and let's get started!

Understanding Integration by Parts

The key to solving ∫xsin(x) dx lies in a technique called integration by parts. This method is essentially the reverse of the product rule for differentiation. Remember that? The product rule states that the derivative 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)

Integration by parts cleverly uses this rule to help us integrate products of functions. The formula for integration by parts is:

∫u dv = uv - ∫v du

Where:

• u is a function we choose to differentiate.
• dv is the remaining part of the integral that we choose to integrate.
• du is the derivative of u.
• v is the integral of dv.

The trick is choosing u and dv wisely. The goal is to select a u that becomes simpler when differentiated, and a dv that is easy to integrate. If you make the right choice, the integral on the right side of the equation (∫v du) will be easier to solve than the original integral (∫u dv).

Why does this work? Think of it as a strategic rearrangement. We're essentially shifting the difficulty from one integral to another, hoping that the new integral is something we can handle more easily. It's like moving furniture in a room to make it more comfortable and functional. The total "stuff" in the room (the overall problem) doesn't change, but its arrangement does, making it easier to navigate.

Choosing u and dv Wisely

For the integral ∫xsin(x) dx, the best choice for u and dv is:

• u = x
• dv = sin(x) dx

Why is this a good choice?

• If we choose u = x, then du = dx. Notice that du is simpler than u; it's just a constant differential. This is exactly what we want!
• If we choose dv = sin(x) dx, then v = -cos(x). Integrating sin(x) is straightforward, and we get -cos(x). The negative sign might seem a bit annoying, but it's nothing we can't handle.

If we had chosen u = sin(x) and dv = x dx, we would have found that du = cos(x) dx and v = (x^2)/2. While this is perfectly valid, the new integral ∫v du would be ∫((x^2)/2)cos(x) dx, which is actually more complicated than the original integral! This is why the initial choice is crucial.

Applying the Integration by Parts Formula

Now that we've chosen our u and dv, let's plug them into the integration by parts formula:

∫u dv = uv - ∫v du

Substituting our choices, we get:

∫xsin(x) dx = x(-cos(x)) - ∫(-cos(x)) dx

Simplify this:

∫xsin(x) dx = -xcos(x) + ∫cos(x) dx

Notice how the integral on the right side is much simpler than the original integral. We've successfully used integration by parts to transform the problem into something easier to solve.

Solving the Remaining Integral

The remaining integral, ∫cos(x) dx, is a standard integral that we should know:

∫cos(x) dx = sin(x) + C

Where C is the constant of integration. Don't forget to include this constant in your final answer! It represents the fact that the derivative of a constant is zero, so when we integrate, we're only finding a family of functions that differ by a constant.

The Final Solution

Now we can substitute this result back into our equation:

∫xsin(x) dx = -xcos(x) + sin(x) + C

And that's it! We've successfully solved the integral of xsin(x) dx using integration by parts. The final answer is:

∫xsin(x) dx = -xcos(x) + sin(x) + C

Let's recap the steps:

1. Identify the integral: ∫xsin(x) dx
2. Choose u and dv: u = x, dv = sin(x) dx
3. Find du and v: du = dx, v = -cos(x)
4. Apply integration by parts formula: ∫u dv = uv - ∫v du
5. Substitute: ∫xsin(x) dx = -xcos(x) - ∫(-cos(x)) dx
6. Simplify: ∫xsin(x) dx = -xcos(x) + ∫cos(x) dx
7. Solve the remaining integral: ∫cos(x) dx = sin(x) + C
8. Substitute back: ∫xsin(x) dx = -xcos(x) + sin(x) + C

Common Mistakes to Avoid

• Forgetting the constant of integration: Always remember to add + C to your final answer when finding indefinite integrals.
• Incorrectly applying the integration by parts formula: Double-check your substitution and make sure you're using the formula correctly.
• Choosing the wrong u and dv: This can lead to a more complicated integral. Practice choosing u and dv to get better at it.
• Sign errors: Be careful with negative signs, especially when integrating trigonometric functions.

Practice Problems

To solidify your understanding, try solving these similar integrals:

1. ∫xcos(x) dx
2. ∫xe^x dx
3. ∫ln(x) dx (Hint: Let u = ln(x) and dv = dx)

Working through these problems will help you master integration by parts and become more confident in your calculus skills.

Conclusion

So there you have it! Solving ∫xsin(x) dx might seem daunting at first, but by using integration by parts and carefully choosing our u and dv, we can break it down into manageable steps. Remember to practice, avoid common mistakes, and always double-check your work. With a little effort, you'll be a pro at integration in no time! Keep up the great work, and happy integrating! And remember, calculus is not as scary as it seems!