Solve the following problems: 1. Find integral 2 theta theta dtheta. 2. Use the matrix method to solve the following simultaneous equations: 3x + 4y = 2, 5x − 6y = 16.

Here are the solutions to the problems:

1. Find $\int \sin^2 \theta \cos \theta \, d\theta$.

Step 1: Use substitution. Let $u = \sin \theta$.

Step 2: Differentiate $u$ with respect to $\theta$: $du = \cos \theta \, d\theta$.

Step 3: Substitute $u$ and $du$ into the integral:

$$\int u^2 \, du$$

Step 4: Integrate with respect to $u$:

$$\frac{u^{2+1}}{2+1} + C = \frac{u^3}{3} + C$$

Step 5: Substitute back $u = \sin \theta$:

$$\frac{\sin^3 \theta}{3} + C$$

The final answer is $\boxed{\frac{\sin^3 \theta}{3} + C}$.

2. Use the matrix method to solve the following simultaneous equations:
$3x + 4y = 2$
$5x - 6y = 16$

Step 1: Write the system in matrix form $AX = B$:

$$\begin{bmatrix} 3 & 4 \\ 5 & -6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2 \\ 16 \end{bmatrix}$$

Step 2: Find the determinant of matrix $A$:

$$\det(A) = (3)(-6) - (4)(5) = -18 - 20 = -38$$

Step 3: Find the inverse of matrix $A$, $A^{-1}$:

$$A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} -6 & -4 \\ -5 & 3 \end{bmatrix} = \frac{1}{-38} \begin{bmatrix} -6 & -4 \\ -5 & 3 \end{bmatrix} = \begin{bmatrix} \frac{3}{19} & \frac{2}{19} \\ \frac{5}{38} & -\frac{3}{38} \end{bmatrix}$$

Step 4: Calculate $X = A^{-1}B$:

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