integrate xsinx

Understand the Problem

The question is asking us to find the integral of the function xsin(x), which involves applying integration techniques such as integration by parts.

Answer

$$ \int x \sin(x) \, dx = -x \cos(x) + \sin(x) + C $$

Steps to Solve

Identify the parts for integration by parts

We will use the integration by parts formula:
$$ \int u , dv = uv - \int v , du $$
For our function $x \sin(x)$, we can let:

$u = x$ (which implies $du = dx$)
$dv = \sin(x) , dx$ (which gives us $v = -\cos(x)$)

Apply the integration by parts formula

Substituting our choices into the integration by parts formula, we have:
$$ \int x \sin(x) , dx = -x \cos(x) - \int -\cos(x) , dx $$
This simplifies to:
$$ \int x \sin(x) , dx = -x \cos(x) + \int \cos(x) , dx $$

Integrate the remaining integral

Now, we need to integrate $\int \cos(x) , dx$, which is:
$$ \int \cos(x) , dx = \sin(x) $$

Combine the results

Putting everything together, we substitute the integral back into our equation:
$$ \int x \sin(x) , dx = -x \cos(x) + \sin(x) + C $$
Where $C$ is the constant of integration.

The integral of the function $x \sin(x)$ is:
$$ \int x \sin(x) , dx = -x \cos(x) + \sin(x) + C $$

More Information

Integration by parts is a common technique used in calculus for integrating products of functions. It's often encountered in problems involving polynomials multiplied by trigonometric, logarithmic, or exponential functions.

Tips

Incorrectly identifying $u$ and $dv$: Make sure to choose parts that simplify the integral correctly.
Forgetting to include the constant of integration $C$: Always remember to add $C$ at the end of the indefinite integral.

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