© Chapter 1 Electric For... HW \# 1 (1) Chapter 2 Electric Fie... Assignment 1 0 $$ + $$ $$ \square $$ $$ \uparrow $$ 2) $$ \int x^{2} \sin (4 x) d x= $$

Question

UpStudy AI · Accepted Answer

To evaluate the integral:

$$
\int x^{2} \sin(4x) \, dx
$$

we'll use the **integration by parts** method twice. Integration by parts is based on the formula:

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

### First Integration by Parts

1. **Choose $$ u $$ and $$ dv $$:**
   $$
   u = x^{2} \quad \Rightarrow \quad du = 2x \, dx
   $$
   $$
   dv = \sin(4x) \, dx \quad \Rightarrow \quad v = -\frac{1}{4} \cos(4x)
   $$
   
2. **Apply the integration by parts formula:**
   $$
   \int x^{2} \sin(4x) \, dx = u \cdot v - \int v \cdot du
   $$
   $$
   = -\frac{x^{2}}{4} \cos(4x) - \int \left(-\frac{1}{4} \cos(4x)\right) \cdot 2x \, dx
   $$
   $$
   = -\frac{x^{2}}{4} \cos(4x) + \frac{1}{2} \int x \cos(4x) \, dx
   $$

### Second Integration by Parts

Now, we need to evaluate the remaining integral:

$$
\int x \cos(4x) \, dx
$$

1. **Choose $$ u $$ and $$ dv $$ for this integral:**
   $$
   u = x \quad \Rightarrow \quad du = dx
   $$
   $$
   dv = \cos(4x) \, dx \quad \Rightarrow \quad v = \frac{1}{4} \sin(4x)
   $$
   
2. **Apply the integration by parts formula again:**
   $$
   \int x \cos(4x) \, dx = u \cdot v - \int v \cdot du
   $$
   $$
   = \frac{x}{4} \sin(4x) - \int \frac{1}{4} \sin(4x) \, dx
   $$
   $$
   = \frac{x}{4} \sin(4x) + \frac{1}{16} \cos(4x) + C
   $$
   (where $$ C $$ is the constant of integration)

### Combine the Results

Substitute the result of the second integration back into the first:

$$
\int x^{2} \sin(4x) \, dx = -\frac{x^{2}}{4} \cos(4x) + \frac{1}{2} \left( \frac{x}{4} \sin(4x) + \frac{1}{16} \cos(4x) \right) + C
$$

Simplify:

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

Combine like terms:

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

### Final Answer

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

where $$ C $$ is the constant of integration.
$$ \int x^{2} \sin(4x) \, dx = \left( -\frac{x^{2}}{4} + \frac{1}{32} \right) \cos(4x) + \frac{x}{8} \sin(4x) + C $$

© Chapter 1 Electric For…
HW # 1 (1)
Chapter 2 Electric Fie…
Assignment 1
0

2)

Upstudy AI Solution

Answer

Solution

First Integration by Parts

Second Integration by Parts

Combine the Results

Final Answer

The Deep Dive

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© Chapter 1 Electric For… HW # 1 (1) Chapter 2 Electric Fie… Assignment 1 0 2)