Calculus

Step 1: Applying integration by parts.
To solve the integral , we use integration by parts. Let: Then: Applying the integration by parts formula , we get:
Step 2: Conclusion.
Thus, the value of simplifies to: Therefore, the correct answer is (3) .

Step 1: Simplify the expression.
We have: Simplifying the expression:
Step 2: Evaluate the limit.
As , approaches 0. Therefore, the limit is:
Step 3: Conclusion.
Therefore, the correct answer is , and the correct option is (3) -1.

Step 1: Continuity condition.
For to be continuous at , the limit of as must be equal to . Therefore, we must compute the limit:
Step 2: Applying L'Hopital's Rule.
This is of the indeterminate form , so we apply L'Hopital's Rule. Differentiating the numerator and denominator: Thus, the limit becomes:
Step 3: Evaluating the limit.
Substitute into the expression: Therefore, for to be continuous at , we must have .
Step 4: Conclusion.
Thus, the value of is . The correct answer is (1) .

Step 1: Differentiating the function.
The function is . To find the critical points, we differentiate : Simplifying: Thus, .
Step 2: Solving for critical points.
Set : This gives two solutions:
Step 3: Second derivative test.
To determine whether these points are maxima or minima, we compute the second derivative: Evaluating at : Evaluating at :
Step 4: Conclusion.
Therefore, the local maximum occurs at and the local minimum occurs at . The correct answer is (1) and .

Step 1: Find the slope of the tangent.
The given curve is . First, find the derivative of : At , the slope of the tangent is:
Step 2: Find the slope of the normal.
The slope of the normal is the negative reciprocal of the tangent slope:
Step 3: Equation of the normal.
Using the point , where , the equation of the normal is: Simplifying: Multiplying through by 2 to remove decimals:
Step 4: Conclusion.
The equation of the normal is . The correct answer is (4) .

Step 1: Slope of the tangent line.
The equation of the line is . The slope of the line is 4. For the tangents on the curve to be parallel to this line, their slopes must also be 4. The derivative of the given function gives the slope of the tangent at any point :
Step 2: Set the derivative equal to the slope of the line.
We set the derivative equal to 4 to find the points where the tangent is parallel to the line: Factoring the quadratic: Thus, and .
Step 3: Find the corresponding -coordinates.
For , substitute into the original equation: So the point is . For , substitute into the original equation: So the point is .
Step 4: Conclusion.
The points on the curve where the tangents are parallel to the line are and . Thus, the correct answer is (3) and .

Step 1: Using the given equation.
We are given that: From the identity , we know that: Therefore, .
Step 2: Using the trigonometric identity.
Using the identity , we have: Squaring both sides, we get:
Step 3: Conclusion.
Thus, . The correct answer is (1) .