Integral

Step 1: Concept
Use the standard integral form: .
Step 2: Analysis
Let . Then . The integrand matches the form .
Step 3: Conclusion
The integral is .
Final Answer: (B)

Step 1: Concept
Multiply numerator and denominator by or add/subtract in the numerator.
Step 2: Analysis
.
Step 3: Conclusion
.
Final Answer: (D)

Step 1: Concept
Use the identity .
Step 2: Analysis
. Now integrate: .
Step 3: Conclusion
.
Final Answer: (A)

Step 1: Concept
Multiply numerator and denominator by .
Step 2: Analysis
. Let , then . .
Step 3: Conclusion
.
Final Answer: (A)

Step 1: Substitution
Put . Also, .
Step 2: Transformation
.
Step 3: Integration
.
Final Answer: (c)

Step 1: Trigonometric Simplification
.
Step 2: Apply Integration Property
Use the rule . Here, and .
Step 3: Conclusion
Result .
Final Answer: (a)

Step 1: Reduction Formula
The standard reduction formula for sine is .
Step 2: Rearrangement
Multiply by : . .
Final Answer: (c)

Step 1: Concept
Use the standard integral form: .
Step 2: Analysis
Let . Then . The integrand matches the form .
Step 3: Conclusion
The integral is .
Final Answer: (B)

Step 1: Substitution to simplify the integral.
We are given the integral: Let us make the substitution . This implies: Therefore, the integral becomes:

Step 2: Recognizing the standard integral form.
We recognize that the integral: is a standard integral whose result is the inverse sine function . Therefore, we have:

Step 3: Substituting back .
Now, substitute back into the equation:

Step 4: Conclusion.
Thus, the value of the integral is: This matches option (D), which is the correct answer.

Step 1: Setting up the partial fractions decomposition.
We are given the integral: We can perform partial fraction decomposition to break this down into simpler fractions. The goal is to express: Multiplying both sides by , we get:

Step 2: Solving for constants and .
Expanding both sides: Equating the coefficients of , we get: Solving this system of equations:

Step 3: Writing the integral.
Now, we substitute these values of and into the partial fractions expression:

Step 4: Integrating each term.
Each of these integrals is a standard inverse tangent form: Thus:

Step 5: Conclusion.
The final solution is: This matches option (A), which is the correct answer.

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}



For real roots:

Step 3: Final Answer

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}



Since ,

Step 3: Final Answer

Concept: Use identity:

Step 1: Simplify denominator.


Step 2: Rewrite integral.


Step 3: Use standard identity.


Step 4: Substitute .
Integral reduces to cotangent form.

Step 5: Final integration.
Conclusion:
Answer = Option (D)

Concept: Use substitution method with derivative matching.

Step 1: Let .

Step 2: Differentiate .


Step 3: Substitute in integral.


Step 4: Integrate.


Step 5: Substitute back.
Conclusion:
Answer = Option (D)

Concept: Split expression and use substitution with trig identities.

Step 1: Rewrite expression.


Step 2: Use substitution .


Step 3: Convert terms.


Step 4: Simplify integral.
Expression reduces to sum of standard forms.

Step 5: Integrate.
Conclusion:
Answer = Option (A)

Concept: Use substitution method:

Step 1: Substitution

Step 2: Transform integral

Step 3: Integrate Final: