Q1. Evaluate by completing the square in .

Background

Topic: Integration using trigonometric substitution and completing the square.

This question tests your ability to manipulate a quadratic expression under a square root and use trigonometric substitution to evaluate the integral.

Key Terms and Formulas

• Completing the square:

• Trigonometric substitution for :

• Inverse trigonometric integrals:

Step-by-Step Guidance

1. Start by rewriting the quadratic under the square root: .

2. Factor out to make the term positive: .

3. Complete the square for to write it in the form .

4. Express the denominator as and set up a substitution .

Try solving on your own before revealing the answer!

Q2. Evaluate using trigonometric identities.

Background

Topic: Integration using trigonometric identities and simplification.

This question tests your ability to manipulate trigonometric expressions and integrate using basic identities.

Key Terms and Formulas

• Quotient of trigonometric functions:

• Integral of :

• Double angle identities:

Step-by-Step Guidance

1. Split the integral into two terms: .

2. Recognize as and as .

3. Integrate each term separately, using substitution if necessary (e.g., ).

Try solving on your own before revealing the answer!

Q3. Evaluate using integration by parts.

Background

Topic: Integration by parts for definite integrals.

This question tests your ability to apply the integration by parts formula to a definite integral.

Key Terms and Formulas

• Integration by parts:

• Derivative and integral of

Step-by-Step Guidance

1. Let and . Compute and .

2. Apply the integration by parts formula: .

3. Evaluate the resulting expression at the bounds and .

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Q4. Evaluate using a trigonometric substitution.

Background

Topic: Integration using trigonometric substitution for rational functions.

This question tests your ability to use substitution to simplify the integral.

Key Terms and Formulas

• Trigonometric substitution: ,

• Identity:

Step-by-Step Guidance

1. Let , so .

2. Substitute into the integral: .

3. Simplify the denominator using the identity .

4. Integrate the resulting expression with respect to .

Try solving on your own before revealing the answer!

Q5. Evaluate using partial fractions.

Background

Topic: Integration using partial fraction decomposition.

This question tests your ability to decompose a rational function and integrate each term.

Key Terms and Formulas

• Partial fractions: Express as a sum of simpler fractions.

• Factor the denominator:

• Integrate terms of the form

Step-by-Step Guidance

1. Factor the denominator: .

2. Set up the partial fraction decomposition: .

3. Solve for and by equating coefficients.

4. Integrate each term separately.

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Q6. Evaluate the improper integral .

Background

Topic: Improper integrals and convergence.

This question tests your ability to evaluate an improper integral with an infinite lower bound.

Key Terms and Formulas

• Improper integral:

• Power rule for integration: (for )

Step-by-Step Guidance

1. Rewrite the integral as a limit: .

2. Integrate with respect to using the power rule.

3. Evaluate the antiderivative at the bounds and .

4. Take the limit as .

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Q7. Evaluate the improper integral .

Background

Topic: Improper integrals and arctangent integration.

This question tests your ability to evaluate an improper integral over the entire real line involving a quadratic denominator.

Key Terms and Formulas

• Improper integral:

• Arctangent integral:

Step-by-Step Guidance

1. Recognize the integral as an arctangent form with .

2. Write the indefinite integral: .

3. Evaluate the definite integral from to using limits.

Try solving on your own before revealing the answer!

Q8. Evaluate the improper integral .

Background

Topic: Improper integrals and substitution.

This question tests your ability to handle an improper integral with a singularity at the lower bound and use substitution to evaluate it.

Key Terms and Formulas

• Substitution: Let , so

• Power rule for integration: (for )

• Improper integral at a bound: Use limits if the integrand is undefined at a bound.

Step-by-Step Guidance

1. Let , so when , ; when , .

2. Rewrite the integral in terms of : .

3. Integrate with respect to using the power rule.

4. Evaluate the result at the bounds and (considering the limit as ).

Try solving on your own before revealing the answer!