Questions: ∫ 3x^2 sin(πx) dx

Solution Steps

To solve the integral \(\int 3 x^{2} \sin (\pi x) d x\), we can use integration by parts. Integration by parts is based on the formula \(\int u \, dv = uv - \int v \, du\). We will choose \(u = x^2\) and \(dv = 3 \sin(\pi x) \, dx\). Then, we will differentiate \(u\) to find \(du\) and integrate \(dv\) to find \(v\). Finally, we will apply the integration by parts formula to find the solution.

We start with the integral we want to solve: \[ \int 3 x^{2} \sin (\pi x) \, dx \]

Using integration by parts, we let:

• \( u = x^2 \) \(\Rightarrow du = 2x \, dx\)
• \( dv = 3 \sin(\pi x) \, dx \) \(\Rightarrow v = -\frac{3}{\pi} \cos(\pi x)\)

Applying the integration by parts formula \(\int u \, dv = uv - \int v \, du\), we have: \[ \int 3 x^{2} \sin (\pi x) \, dx = -\frac{3}{\pi} x^2 \cos(\pi x) - \int -\frac{3}{\pi} \cdot 2x \cos(\pi x) \, dx \]

This simplifies to: \[ -\frac{3}{\pi} x^2 \cos(\pi x) + \frac{6}{\pi} \int x \cos(\pi x) \, dx \] We apply integration by parts again on \(\int x \cos(\pi x) \, dx\) with:

• \( u = x \) \(\Rightarrow du = dx\)
• \( dv = \cos(\pi x) \, dx \) \(\Rightarrow v = \frac{1}{\pi} \sin(\pi x)\)

Thus, we have: \[ \int x \cos(\pi x) \, dx = \frac{1}{\pi} x \sin(\pi x) - \frac{1}{\pi} \int \sin(\pi x) \, dx \] The integral of \(\sin(\pi x)\) is: \[ -\frac{1}{\pi} \cos(\pi x) \] Putting it all together, we find: \[ \int x \cos(\pi x) \, dx = \frac{1}{\pi} x \sin(\pi x) + \frac{1}{\pi^2} \cos(\pi x) \]

Substituting back, we get: \[ \int 3 x^{2} \sin (\pi x) \, dx = -\frac{3}{\pi} x^2 \cos(\pi x) + \frac{6}{\pi} \left( \frac{1}{\pi} x \sin(\pi x) + \frac{1}{\pi^2} \cos(\pi x) \right) \] This simplifies to: \[ -\frac{3}{\pi} x^2 \cos(\pi x) + \frac{6}{\pi^2} x \sin(\pi x) + \frac{6}{\pi^3} \cos(\pi x) \]

Final Answer