What is the general solution to the following: $$ \int 2 x \sin (x) d x $$ ?

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UpStudy AI · Accepted Answer

To find the general solution to the integral:

$$
\int 2x \sin(x) \, dx
$$

we can use the **integration by parts** method. Integration by parts is based on the formula:

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

**Step 1: Choose $$ u $$ and $$ dv $$**

Let:
$$
u = 2x \quad \Rightarrow \quad du = 2 \, dx
$$
$$
dv = \sin(x) \, dx \quad \Rightarrow \quad v = -\cos(x)
$$

**Step 2: Apply the Integration by Parts Formula**

$$
\int 2x \sin(x) \, dx = u \cdot v - \int v \, du
$$
$$
= (2x)(-\cos(x)) - \int (-\cos(x))(2) \, dx
$$
$$
= -2x \cos(x) + 2 \int \cos(x) \, dx
$$

**Step 3: Integrate $$\cos(x)$$**

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

**Step 4: Combine the Results**

$$
-2x \cos(x) + 2 \sin(x) + C
$$

where $$ C $$ is the constant of integration.

**Final Answer:**

$$
\boxed{-2x\,\cos x\,+\,2\,\sin x\,+\,C}
$$
The general solution to the integral $$ \int 2x \sin(x) \, dx $$ is: $$ -2x \cos x + 2 \sin x + C $$ where $$ C $$ is the constant of integration.

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