Integral Calculus

Step 1: Apply the chain rule.
The derivative of w.r.t. is given by: and
Step 2: Evaluate at .
At , and . So, substituting the given values: Thus,
Step 3: Conclusion.
The correct answer is .

Step 1: Use the property of definite integrals.
Let Using the property

Step 2: Replace by .


Step 3: Add both expressions.


Step 4: Use the identity .


Step 5: Conclusion.

Step 1: Use partial fractions.
We are given the integral We can decompose the integrand using partial fractions:
Step 2: Solve for constants.
Multiplying both sides by , we get: Solving for and , we find the constants. After simplifying, we can integrate the resulting terms.
Step 3: Integration.
Using the standard integrals for terms of the form , the final result of the integral is . Thus, the correct answer is option (B).

Step 1: Simplify the integrand.
We begin by recognizing the symmetry in the integral. By simplifying the expression using trigonometric identities, we can find that the integral evaluates to .
Step 2: Conclusion.
Thus, the value of the integral is , corresponding to option (B).

Step 1: Simplify the integrand.
We are asked to integrate . To handle this, first simplify the denominator and make a substitution to simplify the integral.
Step 2: Use the standard integral.
This is a standard integral and can be integrated using the formula for the integral of rational functions with square roots. After applying the formula, we get:
Step 3: Conclusion.
Thus, the value of the integral is , corresponding to option (B).

Step 1: Understanding the integral.
We are asked to evaluate the integral To solve this, we recognize that it is a rational function in terms of , and simplifying this expression will make the integration easier.
Step 2: Simplifying the expression.
Let . Then, . The integral becomes: The result of the integration is:
Step 3: Conclusion.
Thus, the integral evaluates to , corresponding to option (D).

Step 1: Use Beta function property.
The integral is of the form
Step 2: Identify values.
Here , .

Step 3: Substitute values.

Step 1: Understanding the integral.
The given integral has the form , which is a standard trigonometric integral. We recognize that we can simplify it using the inverse sine function.

Step 2: Simplifying the expression.
Rearranging the terms inside the square root, we get the expression in the form , which corresponds to the standard formula for this type of integral.

Step 3: Conclusion.
Thus, the correct answer is option (A):

Step 1: Use the identity for .


Step 2: Substitute into the integral.


Step 3: Rewrite using tangent and secant.


Step 4: Integrate.


Step 5: Conclusion.

Step 1: Simplify the integrand.
The integrand can be simplified by recognizing that can be rewritten using trigonometric identities.
Step 2: Use substitution.
Substitute and simplify the integral, then apply the standard integral formula to get the result .
Step 3: Conclusion.
The correct answer is (B) .

Step 1: Applying the Mean Value Theorem.
The Mean Value Theorem (MVT) states that for a function continuous on and differentiable on , there exists some such that: For this problem, , and we apply the MVT on the interval .

Step 2: Finding .
The derivative of is: Thus, . By applying the MVT, we find that .

Step 3: Conclusion.
Thus, the value of is , which makes option (A) the correct answer.

Step 1: Simplify the integrand.
We start by simplifying the integrand: This is a standard form that can be simplified as:
Step 2: Recognize the symmetry.
The integrand is an odd function, because . The integral of an odd function over a symmetric interval is 0. Thus, the value of the integral is:
Step 3: Conclusion.
Thus, the value of the integral is 0, which corresponds to option (A).

Step 1: Simplify the integral.
We are given the integral: We know that , so the denominator becomes: Thus, the integral simplifies to:
Step 2: Solve the integral.
The integral is simplified and computed, giving the result , corresponding to option (B).

Step 1: Use substitution.
Let . Then
Step 2: Substitute into the integral.

Step 3: Integrate.

Step 4: Substitute back.

Step 1: Divide numerator and denominator by .


Step 2: Use substitution.
Let

Step 3: Simplify the denominator.


Step 4: Integrate.


Step 5: Back substitute.

Step 1: Use the property of definite integrals.
Let Using the property .

Step 2: Replace by .


Step 3: Add both expressions.


Step 4: Conclusion.

Step 1: Use the identity for .

Step 2: Substitute into the integral.

Step 3: Integrate termwise.

Step 4: Apply limits.

Step 1: Use substitution.
Let , then , so the integral becomes:
Step 2: Integrate.
Now, integrate . Substituting back , we get:
Step 3: Conclusion.
The correct answer is (B) .

Step 1: Simplify the integrand.
We are given the integral: We can rewrite the expression as , and then simplify the integral to:
Step 2: Use substitution.
We know that is the inverse of , so we can make a substitution: Substituting into the integral, we get:
Step 3: Conclusion.
Thus, the solution is , which corresponds to option (B).

Step 1: Simplify the integrand.
We are asked to evaluate the integral We can simplify the expression by using the identity , so the integral becomes:
Step 2: Use symmetry.
The integrand has symmetry, and after applying suitable trigonometric identities and simplifications, we get the result for the integral as .

Step 3: Conclusion.
Thus, the value of the integral is , which corresponds to option (B).

Step 1: Break the integral into intervals based on the greatest integer function.
The greatest integer function takes integer values between the intervals where is between integer values. Break the integral into separate intervals: - From to , - From to , - From to , Step 2: Calculate the integral.
Now calculate each part of the integral: So the total integral is: Step 3: Conclusion.
The correct answer is (B) 0.

Step 1: Express in terms of .
From , The Y-axis corresponds to .

Step 2: Set up the area integral.
Required area

Step 3: Integrate.


Step 4: Multiply by 2 (both sides of Y-axis symmetry).


Step 5: Conclusion.

Step 1: Understanding the logical implication.
For the implication to be false, the antecedent must be true and the consequent must be false. Therefore, and . The only way can be false is if and .

Step 2: Conclusion.
Thus, the truth values of and are and , which makes option (D) the correct answer.

Step 1: Simplify the integral.
We are given the integral . Use substitution and properties of exponential functions to simplify the expression. Step 2: Apply integration.
After performing the integration using the appropriate substitution method, we find that: Step 3: Conclusion.
The correct answer is (C) .

Step 1: Choose an appropriate substitution.
Let

Step 2: Differentiate with respect to .


Step 3: Substitute into the integral.


Step 4: Integrate and back-substitute.


Step 5: Conclusion.

Step 1: Identify the nature of the function.
Consider the integrand Here, is an even function, while is an odd function because replacing by changes the sign of the numerator but not the denominator.

Step 2: Determine the parity of the integrand.
The product of an even function and an odd function is an odd function. Hence, is an odd function.

Step 3: Use the property of definite integrals.
For any odd function , Therefore,

Step 4: Final conclusion.
Using the symmetry property of odd functions over symmetric limits, the value of the given integral is zero.

Step 1: Use partial fractions.

Step 2: Integrate term by term.

Step 3: Evaluate the definite integral.

Step 1: Simplify the integrand.
We are asked to evaluate the integral: We can use the identity to express in terms of . This simplifies the integrand and allows us to rewrite the expression in a more manageable form.
Step 2: Apply a substitution.
We then use the substitution , so , to transform the integral into a standard form. After solving the integral, we get the result:
Step 3: Conclusion.
Thus, the solution to the integral is , which corresponds to option (D).

Step 1: Set up the integral for the area.
To find the area, we need to integrate the given function from 0 to the point where it intersects the x-axis. First, solve for the x-intercepts of the curve by setting . Thus, the x-intercepts are at and .
Step 2: Integrate the function.
Now, integrate the function from 0 to 4: Evaluating the integral, we get the area as square units.
Step 3: Conclusion.
Thus, the area of the region is , corresponding to option (B).

Step 1: Use substitution.
Let . Then, , and the integral becomes:
Step 2: Solve the integral.
The integral simplifies to:
Step 3: Conclusion.
Thus, the integral evaluates to , corresponding to option (D).

Step 1: Apply integration techniques.
We use the substitution , so . The limits change accordingly, and we integrate the resulting expression: Step 2: Simplify the result.
After evaluating the integral, we find: Step 3: Conclusion.
The correct answer is (B) .

Step 1: Rewrite the integrand.
We are given the integral Split the integral into two parts: Step 2: Use the identity for tangent.
Recall that Thus, the second term becomes Step 3: Apply differentiation insight.
Observe that Hence, the given integrand matches the derivative structure of Step 4: Write the final result.
Therefore, the integral evaluates to

Step 1: Rewrite the given integral:
The integral is:

Step 2: Use substitution:
Let: Using the trigonometric identities: The limits of integration change as follows:

Step 3: Substitute into the integral:
Substituting and into the integral:
Simplify the expression:
Combine terms:
Simplify the powers:

Step 4: Use a new substitution:
Let: The limits of integration change: Substitute into the integral: Combine terms:

Step 5: Simplify the integral:
Let: The limits change again:
Substitute into the integral:
Step 6: Solve the simplified integral:
The integral of is: Evaluate the definite integral: Substitute the limits: Simplify: Rewrite using exponents: Factorize:

Final Answer:

Step 1: Observe the integral and try substitution. Let: Step 2: Now, the integral becomes: Thus, the solution is:

The surface area of a sphere is given by: where is the radius of the sphere.
The volume of the sphere is given by:
We are given:
We need to find , the rate of increase of volume.
Step 1: Relating and
Differentiate with respect to :
Rearrange to solve for :
Substitute and :
Step 2: Relating and
Differentiate with respect to :
Substitute and :
Simplify:
Final Answer:

Step 1: Formula for Area
Area .
Step 2: Compute Cross Product
.
Step 3: Equate Area
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Final Answer: (A)

Step 1: Substitution
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Step 2: Equate Coefficients
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Step 3: Solve for
.
Final Answer: (B)

Step 1: Truth Values
- : . - : .
Step 2: Contrapositive
A statement and its contrapositive always have the SAME truth value.
Step 3: Conclusion
and .
Final Answer: (C)

Step 1: Total and Defective
Total oranges = 20. Defective (D) = 4, Good (G) = 16. Sample size = 3.
Step 2: Probabilities
- .
- .
Step 3: Conclusion
Based on hypergeometric calculations, the values in the table for option A or B in the paper match these probabilities.
Final Answer: (A)

Concept:
Plane parallel to XY-plane has equation: Step 1: Use given point. Point A =
Step 2: Substitute.
Step 3: Conclusion. Equation of plane is