The given integral is:

$\int x \cos x \, dx$

This is a standard integral that can be solved using integration by parts. The formula for integration by parts is:

$\int u \, dv = uv - \int v \, du$

Step 1: Choose $u$ and $dv$

• Let $u = x$ (since $x$ is algebraic and simplifies upon differentiation).
• Let $dv = \cos x \, dx$.

Now, differentiate $u$ and integrate $dv$:

• $du = dx$ (derivative of $x$).
• $v = \int \cos x \, dx = \sin x$.

Step 2: Apply the formula

Using the formula $\int u \, dv = uv - \int v \, du$, we substitute:

$\int x \cos x \, dx = x \sin x - \int \sin x \, dx$

Step 3: Simplify

We now solve $\int \sin x \, dx$:

$\int \sin x \, dx = -\cos x$

Substitute back into the equation:

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

Simplify:

$\int x \cos x \, dx = x \sin x + \cos x + C$

Final Answer:

The solution to the integral is:

$\int x \cos x \, dx = x \sin x + \cos x + C$

Where $C$ is the constant of integration.

Would you like a detailed explanation on integration by parts or other similar examples? Let me know!

Related Questions:

1. Solve $\int x \sin x \, dx$.
2. Solve $\int x e^x \, dx$.
3. Solve $\int x^2 \cos x \, dx$.
4. Solve $\int x \ln x \, dx$.
5. Prove the formula for integration by parts.

Tip:

When solving integration by parts, always choose $u$ as the term that simplifies upon differentiation.