QUESTION 2 Determine $$ \int y d x $$ if: $$ 2.1 \quad y=\left(x^{2}-2 x+1\right) e^{2 x} $$ $$ 2.2 \quad y=\sin ^{3} 4 x \cos ^{2} 4 x $$

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Let's solve the integrals step by step.

### 2.1: $$ y = (x^{2} - 2x + 1)e^{2x} $$

We need to evaluate the integral:

$$
\int y \, dx = \int (x^{2} - 2x + 1)e^{2x} \, dx
$$

### 2.2: $$ y = \sin^{3}(4x) \cos^{2}(4x) $$

We need to evaluate the integral:

$$
\int y \, dx = \int \sin^{3}(4x) \cos^{2}(4x) \, dx
$$

Now, I will calculate both integrals simultaneously.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \sin^{3}\left(4x\right)\cos^{2}\left(4x\right) dx$$
- step1: Evaluate the integral:
  $$-\frac{\sin^{3-1}\left(4x\right)\cos^{2+1}\left(4x\right)}{4\left(2+3\right)}+\frac{3-1}{2+3}\times \int \sin^{3-2}\left(4x\right)\cos^{2}\left(4x\right) dx$$
- step2: Subtract the numbers:
  $$-\frac{\sin^{2}\left(4x\right)\cos^{2+1}\left(4x\right)}{4\left(2+3\right)}+\frac{3-1}{2+3}\times \int \sin^{3-2}\left(4x\right)\cos^{2}\left(4x\right) dx$$
- step3: Add the numbers:
  $$-\frac{\sin^{2}\left(4x\right)\cos^{3}\left(4x\right)}{4\left(2+3\right)}+\frac{3-1}{2+3}\times \int \sin^{3-2}\left(4x\right)\cos^{2}\left(4x\right) dx$$
- step4: Subtract the numbers:
  $$-\frac{\sin^{2}\left(4x\right)\cos^{3}\left(4x\right)}{4\left(2+3\right)}+\frac{3-1}{2+3}\times \int \sin^{1}\left(4x\right)\cos^{2}\left(4x\right) dx$$
- step5: Add the numbers:
  $$-\frac{\sin^{2}\left(4x\right)\cos^{3}\left(4x\right)}{4\times 5}+\frac{3-1}{2+3}\times \int \sin^{1}\left(4x\right)\cos^{2}\left(4x\right) dx$$
- step6: Evaluate the power:
  $$-\frac{\sin^{2}\left(4x\right)\cos^{3}\left(4x\right)}{4\times 5}+\frac{3-1}{2+3}\times \int \sin\left(4x\right)\cos^{2}\left(4x\right) dx$$
- step7: Multiply the numbers:
  $$-\frac{\sin^{2}\left(4x\right)\cos^{3}\left(4x\right)}{20}+\frac{3-1}{2+3}\times \int \sin\left(4x\right)\cos^{2}\left(4x\right) dx$$
- step8: Subtract the numbers:
  $$-\frac{\sin^{2}\left(4x\right)\cos^{3}\left(4x\right)}{20}+\frac{2}{2+3}\times \int \sin\left(4x\right)\cos^{2}\left(4x\right) dx$$
- step9: Add the numbers:
  $$-\frac{\sin^{2}\left(4x\right)\cos^{3}\left(4x\right)}{20}+\frac{2}{5}\times \int \sin\left(4x\right)\cos^{2}\left(4x\right) dx$$
- step10: Simplify the expression:
  $$-\frac{1}{20}\sin^{2}\left(4x\right)\cos^{3}\left(4x\right)+\frac{2}{5}\times \int \sin\left(4x\right)\cos^{2}\left(4x\right) dx$$
- step11: Evaluate the integral:
  $$-\frac{1}{20}\sin^{2}\left(4x\right)\cos^{3}\left(4x\right)+\frac{2}{5}\left(-\frac{1}{12}\cos^{3}\left(4x\right)\right)$$
- step12: Multiply the terms:
  $$-\frac{1}{20}\sin^{2}\left(4x\right)\cos^{3}\left(4x\right)-\frac{1}{30}\cos^{3}\left(4x\right)$$
- step13: Add the constant of integral C:
  $$-\frac{1}{20}\sin^{2}\left(4x\right)\cos^{3}\left(4x\right)-\frac{1}{30}\cos^{3}\left(4x\right) + C, C \in \mathbb{R}$$
Calculate the integral $$ (x^{2} - 2*x + 1)*e^{2*x} $$.
Evaluate the integral by following steps:
- step0: Evaluate using partial integration formula:
  $$\int \left(x^{2}-2x+1\right)e^{2x} dx$$
- step1: Prepare for integration by parts:
  $$\begin{align}&u=x^{2}-2x+1\&dv=e^{2x}dx\end{align}$$
- step2: Calculate the derivative:
  $$\begin{align}&du=\left(2x-2\right)dx\&dv=e^{2x}dx\end{align}$$
- step3: Evaluate the integral:
  $$\begin{align}&du=\left(2x-2\right)dx\&v=\frac{e^{2x}}{2}\end{align}$$
- step4: Substitute the values into formula:
  $$\left(x^{2}-2x+1\right)\times \frac{e^{2x}}{2}-\int \left(2x-2\right)\times \frac{e^{2x}}{2} dx$$
- step5: Calculate:
  $$\frac{\left(x^{2}-2x+1\right)e^{2x}}{2}-\int \left(x-1\right)e^{2x} dx$$
- step6: Evaluate the integral:
  $$\frac{\left(x^{2}-2x+1\right)e^{2x}}{2}-\frac{\left(x-1\right)e^{2x}}{2}+\frac{e^{2x}}{4}$$
- step7: Calculate:
  $$\frac{\left(x^{2}-2x+1\right)e^{2x}}{2}-\frac{xe^{2x}-e^{2x}}{2}+\frac{e^{2x}}{4}$$
- step8: Simplify the expression:
  $$\frac{e^{2x}x^{2}}{2}-xe^{2x}+\frac{e^{2x}}{2}-\frac{xe^{2x}-e^{2x}}{2}+\frac{e^{2x}}{4}$$
- step9: Simplify the expression:
  $$\frac{e^{2x}x^{2}}{2}-xe^{2x}+\frac{e^{2x}}{2}-\left(\frac{e^{2x}x}{2}-\frac{e^{2x}}{2}\right)+\frac{e^{2x}}{4}$$
- step10: Remove the parentheses:
  $$\frac{e^{2x}x^{2}}{2}-xe^{2x}+\frac{e^{2x}}{2}-\frac{e^{2x}x}{2}+\frac{e^{2x}}{2}+\frac{e^{2x}}{4}$$
- step11: Add the constant of integral C:
  $$\frac{e^{2x}x^{2}}{2}-xe^{2x}+\frac{e^{2x}}{2}-\frac{e^{2x}x}{2}+\frac{e^{2x}}{2}+\frac{e^{2x}}{4} + C, C \in \mathbb{R}$$
### Results

#### 2.1: Integral of $$ y = (x^{2} - 2x + 1)e^{2x} $$

The integral is:

$$
\int (x^{2} - 2x + 1)e^{2x} \, dx = \frac{e^{2x}x^{2}}{2} - xe^{2x} + \frac{e^{2x}}{2} - \frac{e^{2x}x}{2} + \frac{e^{2x}}{2} + \frac{e^{2x}}{4} + C, \quad C \in \mathbb{R}
$$

#### 2.2: Integral of $$ y = \sin^{3}(4x) \cos^{2}(4x) $$

The integral is:

$$
\int \sin^{3}(4x) \cos^{2}(4x) \, dx = -\frac{1}{20}\sin^{2}(4x)\cos^{3}(4x) - \frac{1}{30}\cos^{3}(4x) + C, \quad C \in \mathbb{R}
$$

These are the results for both integrals.
The integrals are: 1. $$ \int (x^{2} - 2x + 1)e^{2x} \, dx = \frac{e^{2x}x^{2}}{2} - xe^{2x} + \frac{e^{2x}}{2} - \frac{e^{2x}x}{2} + \frac{e^{2x}}{2} + \frac{e^{2x}}{4} + C $$ 2. $$ \int \sin^{3}(4x) \cos^{2}(4x) \, dx = -\frac{1}{20}\sin^{2}(4x)\cos^{3}(4x) - \frac{1}{30}\cos^{3}(4x) + C $$

QUESTION 2
Determine if:

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