1. Integrate the following by using tabular integration: an = (2/1) ∫ x^2(1 - x)cos(nπx) dx. 2. By using trigonometric substitution, integrate ∫ √(1 - x^2) dx. 3. Find all minors o... 1. Integrate the following by using tabular integration: an = (2/1) ∫ x^2(1 - x)cos(nπx) dx. 2. By using trigonometric substitution, integrate ∫ √(1 - x^2) dx. 3. Find all minors of the matrix A = [3 4 -1; 1 0 3; 2 -5 -4]. Read more

Steps to Solve

1. Tabular Integration Setup

For the integral $a_n = \frac{2}{1} \int x^2 (1 - x) \cos(n \pi x) , dx$, we identify parts for integration by parts (IBP):

Let:

• $u = x^2 (1 - x) \Rightarrow du = (2x - 3x^2) , dx$
• $dv = \cos(n \pi x) , dx \Rightarrow v = \frac{1}{n \pi} \sin(n \pi x)$

2. Applying Integration by Parts

Using the integration by parts formula $\int u , dv = uv - \int v , du$:

Calculate: $$ \int x^2(1 - x) \cos(n \pi x) , dx = \frac{x^2(1 - x)}{n \pi} \sin(n \pi x) - \int \frac{\sin(n \pi x)}{n \pi} (2x - 3x^2) , dx $$

3. Integrate Again as Necessary

Continue integrating the remaining integral, applying IBP as needed, until reaching simpler integrals.

4. Trigonometric Substitution

For $\int \sqrt{1 - x^2} , dx$, we use the substitution $x = \sin(\theta) \Rightarrow dx = \cos(\theta) , d\theta$.

Thus, the integral becomes: $$ \int \sqrt{1 - \sin^2(\theta)} \cos(\theta) , d\theta = \int \cos^2(\theta) , d\theta $$

5. Integrate Using Trigonometric Identity

Utilize the identity $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$: $$ \int \cos^2(\theta) , d\theta = \frac{1}{2} \left( \theta + \frac{1}{2} \sin(2\theta) \right) + C $$

Convert back to $x$ to express the result in terms of $x$.

6. Finding Minors of the Matrix

Given matrix $A = \begin{bmatrix} 3 & 4 & -1 \ 1 & 0 & 3 \ 2 & 5 & -4 \end{bmatrix}$, find minors for each element.

The minor for element $a_{ij}$ is calculated as the determinant of the submatrix obtained by removing the $i^{th}$ row and $j^{th}$ column.

7. Calculate Determinants for Minors

For example, the minor $M_{11}$ (removing the first row and first column): $$ M_{11} = \begin{vmatrix} 0 & 3 \ 5 & -4 \end{vmatrix} = (0)(-4) - (3)(5) = -15 $$

Repeat to find all other minors.