Calculus

Step 1: Understanding the Concept:
The integral represents the area of a quarter-circle of radius 3 in the first quadrant. It can be solved using the standard formula for .

Step 2: Key Formula or Approach:
Formula: .

Step 3: Detailed Explanation:
Here .
Substitute the upper limit :
Substitute the lower limit :
Final result: .

Step 4: Final Answer:
The value of the integral is .

Step 1: Understanding the Concept:
This involves inverse trigonometric identities. Specifically, we must be careful with when , as it introduces a term.

Step 2: Key Formula or Approach:
1. .
2. If , then .
3. If , then .

Step 3: Detailed Explanation:
LHS .
We know .
Now evaluate . Here .
Substitute back into LHS:
We are given: .
Since :
Comparing: .

Step 4: Final Answer:
The value of is .

Step 1: Understanding the Concept:
This is a definite integral problem where the result of the integration is given as a constant. We need to evaluate the integral in terms of the upper limit and solve the resulting quadratic equation.

Step 2: Key Formula or Approach:
1. Power rule of integration: .
2. Fundamental Theorem of Calculus: .
3. Sum of roots of a quadratic equation is .

Step 3: Detailed Explanation:
First, evaluate the integral:
Applying the limits:
Given that the integral equals 5:
This is a quadratic equation in . We need to find the sum of all possible values of .
By comparing with , we have and .
Sum of roots (values of ) .

Step 4: Final Answer:
The sum of all values of is .

Step 1: Understanding the Question:
The integrand is a proper rational function. We should use partial fraction decomposition to integrate it.

Step 2: Key Formula or Approach:
Factor the denominator: .
Let .

Step 3: Detailed Explanation:
Finding constants and :

Put :

Put :

Substituting back into the integral:

Integrating term by term:


Step 4: Final Answer:
The value of the integral is .

Step 1: Understanding the Concept:
The point of intersection of a pair of straight lines is the point where the partial derivatives and vanish.

Step 2: Key Formula or Approach:
1. Condition for pair of lines: .
2. Point of intersection: Solve and .

Step 3: Detailed Explanation:
Let .
First, find using the condition :
.
Now find the intersection point using partial derivatives:
.
.
Substituting :
.
The point is .

Step 4: Final Answer:
The point of intersection is .

Step 1: Understanding the Concept:
When the degree of the numerator is greater than or equal to the degree of the denominator, we use algebraic manipulation or division.

Step 2: Key Formula or Approach:
1. Addition and subtraction in the numerator: .
2. .

Step 3: Detailed Explanation:
Rewrite the expression inside the integral:
Integrate:

Step 4: Final Answer:
The integral is .

Step 1: Understanding the Concept:
We use differentials to find the approximate value. For a function , the value at is .

Step 2: Key Formula or Approach:
Let .
We choose (nearest perfect cube) and .

Step 3: Detailed Explanation:
1. Find :
.
2. Find the derivative :
.
3. Evaluate :
.
4. Calculate the approximation:
.
.
Since :
.

Step 4: Final Answer:
The approximate value is .

Step 1: Understanding the Question:
Eigenvalues of a matrix are the roots of the characteristic equation .

Step 2: Key Formula or Approach:
Set up the characteristic determinant and solve for .

Step 3: Detailed Explanation:
The characteristic equation is:

Expanding the determinant along the first column:



This gives:
or .
Solving for , we get .

Step 4: Final Answer:
The eigenvalues are .