To solve the integral $\int x^2 \sin(4x) \, dx$, we will use integration by parts. Since we have a product of two functions, $x^2$ and $\sin(4x)$, we will apply the technique twice (or use it iteratively).

The formula for integration by parts is:

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

Step 1: First Integration by Parts

Let's choose $u = x^2$ and $dv = \sin(4x) dx$.

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

• $du = 2x \, dx$
• To integrate $dv = \sin(4x) dx$, we find $v$: $\int \sin(4x) dx = -\frac{1}{4} \cos(4x)$

So, the first integration by parts gives: $\int x^2 \sin(4x) \, dx = -\frac{1}{4} x^2 \cos(4x) + \int \frac{1}{4} \cdot 2x \cos(4x) \, dx$

This simplifies to: $\int x^2 \sin(4x) \, dx = -\frac{1}{4} x^2 \cos(4x) + \frac{1}{2} \int x \cos(4x) \, dx$

Step 2: Second Integration by Parts

Now, we need to solve $\int x \cos(4x) \, dx$, so we apply integration by parts again.

Let $u = x$ and $dv = \cos(4x) dx$:

• $du = dx$
• To integrate $dv = \cos(4x) dx$, we find $v$: $\int \cos(4x) dx = \frac{1}{4} \sin(4x)$

So the second integration by parts gives: $\int x \cos(4x) \, dx = \frac{1}{4} x \sin(4x) - \int \frac{1}{4} \sin(4x) \, dx$

Step 3: Simplifying the Remaining Integral

Now, integrate $\int \sin(4x) \, dx$: $\int \sin(4x) \, dx = -\frac{1}{4} \cos(4x)$

Substituting everything back, we get: $\int x \cos(4x) \, dx = \frac{1}{4} x \sin(4x) + \frac{1}{16} \cos(4x)$

Step 4: Putting Everything Together

Now substitute this back into our original expression: $\int x^2 \sin(4x) \, dx = -\frac{1}{4} x^2 \cos(4x) + \frac{1}{2} \left( \frac{1}{4} x \sin(4x) + \frac{1}{16} \cos(4x) \right)$

Simplifying further: $\int x^2 \sin(4x) \, dx = -\frac{1}{4} x^2 \cos(4x) + \frac{1}{8} x \sin(4x) + \frac{1}{32} \cos(4x) + C$

Where $C$ is the constant of integration.

Final Answer:

$\int x^2 \sin(4x) \, dx = -\frac{1}{4} x^2 \cos(4x) + \frac{1}{8} x \sin(4x) + \frac{1}{32} \cos(4x) + C$

Would you like further details or have any questions?

Here are some related questions for you:

1. How do you determine when to apply integration by parts?
2. What if the function involves higher powers of $x$ or other trigonometric functions?
3. How do we apply integration by parts for definite integrals?
4. Can you use substitution instead of integration by parts in this case?
5. What are some common mistakes in integration by parts?

Tip: When dealing with products of polynomials and trigonometric functions, always look for patterns that suggest integration by parts.