Evaluate the integral of x*cos(x) dx
Understand the Problem
The question asks to evaluate the indefinite integral of x*cos(x) with respect to x. This will likely require integration by parts.
Answer
$\int x \cos(x) \, dx = x \sin(x) + \cos(x) + C$
Answer for screen readers
$\int x \cos(x) , dx = x \sin(x) + \cos(x) + C$
Steps to Solve
- Identify $u$ and $dv$
We use integration by parts, which states $\int u , dv = uv - \int v , du$. We need to choose $u$ and $dv$. Let $u = x$ and $dv = \cos(x) , dx$. The choice of $u$ is guided by the fact that differentiation simplifies $x$, and $dv$ must be something we can readily integrate.
- Find $du$ and $v$
Now, differentiate $u$ to find $du$, and integrate $dv$ to find $v$.
$du = dx$ $v = \int \cos(x) , dx = \sin(x)$
- Apply the integration by parts formula
Substitute $u$, $du$, $v$, and $dv$ into the integration by parts formula: $\int u , dv = uv - \int v , du$
$\int x \cos(x) , dx = x \sin(x) - \int \sin(x) , dx$
- Evaluate the remaining integral
Evaluate $\int \sin(x) , dx$:
$\int \sin(x) , dx = -\cos(x)$
- Substitute and simplify
Substitute the result back into the equation:
$\int x \cos(x) , dx = x \sin(x) - (-\cos(x)) + C$ $\int x \cos(x) , dx = x \sin(x) + \cos(x) + C$
$\int x \cos(x) , dx = x \sin(x) + \cos(x) + C$
More Information
The integration by parts technique is essential for integrals involving products of functions, especially when simple substitution doesn't work. In this case, it elegantly simplifies the integral of $x \cos(x)$.
Tips
A common mistake is to incorrectly choose $u$ and $dv$. Choosing $u = \cos(x)$ and $dv = x , dx$ would lead to a more complex integral after applying integration by parts. Another error is forgetting the constant of integration, $C$. Failing to correctly integrate $\sin(x)$ as $-\cos(x)$ is also a frequent mistake.
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