Integrate x cos x
Understand the Problem
The question is asking for the integral of the function x multiplied by the cosine of x. This requires applying integration techniques such as integration by parts.
Answer
$x \sin(x) + \cos(x) + C$
Answer for screen readers
The final answer is $$x \sin(x) + \cos(x) + C$$
Steps to Solve
- Identify components for integration by parts
Integration by parts formula is: $$\int u \ dv = uv - \int v \ du$$
For the integral $$\int x \cos(x) , dx$$, let: $u = x$ and $dv = \cos(x) , dx$
- Differentiate and integrate components
- Differentiate $u = x$: $$\frac{du}{dx} = 1 \implies du = dx$$
- Integrate $dv = \cos(x) , dx$: $$v = \int \cos(x) , dx = \sin(x)$$
- Substitute into the integration by parts formula
Substitute $u$, $v$, $du$, and $dv$ into the integration by parts formula: $$\int x \cos(x) , dx = x \sin(x) - \int \sin(x) , dx$$
- Evaluate the remaining integral
The remaining integral is: $$- \int \sin(x) , dx$$
We know: $$\int \sin(x) , dx = -\cos(x)$$
Therefore: $$- \int \sin(x) , dx = \cos(x)$$
- Combine results
Combine the results to obtain the final answer: $$\int x \cos(x) , dx = x \sin(x) + \cos(x) + C$$
where $C$ is the constant of integration.
The final answer is $$x \sin(x) + \cos(x) + C$$
More Information
Integration by parts is a powerful technique that can be used to solve integrals involving products of functions.
Tips
A common mistake is incorrectly choosing $u$ and $dv$. Make sure to pick them such that the resulting integral is simpler to solve.
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