Question
What is the general solution to the following: \( \int 2 x \sin (x) d x \) ?
Ask by Hill Medina. in the United States
Dec 31,2024
Real Tutor Solution
Tutor-Verified Answer
Answer
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.
Solution
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:**
\[
\boxed{-2x\,\cos x\,+\,2\,\sin x\,+\,C}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To solve the integral \( \int 2 x \sin (x) d x \), we can use integration by parts. Here, we can let \( u = 2x \) and \( dv = \sin(x) dx \). Then, differentiating \( u \) gives us \( du = 2 dx \), and integrating \( dv \) gives us \( v = -\cos(x) \). Applying the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we get: \[ \int 2x \sin(x) dx = -2x \cos(x) - \int -\cos(x) (2 dx) = -2x \cos(x) + 2 \int \cos(x) dx \] The integral of \( \cos(x) \) is \( \sin(x) \), leading us to: \[ \int 2x \sin(x) dx = -2x \cos(x) + 2 \sin(x) + C \] where \( C \) is the constant of integration. Thus, the general solution is: \[ \int 2 x \sin (x) d x = -2x \cos(x) + 2 \sin(x) + C \]
