Q1. Integrate by parts:

Background

Topic: Integration by Parts

This question tests your ability to use the integration by parts technique, which is useful when integrating the product of two functions.

Key Terms and Formulas

• Integration by Parts Formula:

• Choose: and from the integrand, then compute and .

Step-by-Step Guidance

1. Identify and in the integrand. A good choice is and .

2. Compute by differentiating with respect to .

3. Integrate to find .

4. Apply the integration by parts formula: .

5. After applying the formula, you will have a new integral to solve. Set up this new integral, but do not solve it yet.

Try solving on your own before revealing the answer!

Q2. Integrate by parts:

Background

Topic: Integration by Parts

This question asks you to integrate the inverse tangent function, which is best approached using integration by parts.

Key Terms and Formulas

• Integration by Parts Formula:

• Recall that .

Step-by-Step Guidance

1. Let and .

2. Compute by differentiating with respect to .

3. Integrate to find .

4. Apply the integration by parts formula: .

5. Set up the remaining integral, which involves . Do not solve this last integral yet.

Try solving on your own before revealing the answer!

Q3. Integrate by parts:

Background

Topic: Integration by Parts (Repeated/Tabular Integration)

This question involves integrating the product of an exponential and a trigonometric function, which often requires applying integration by parts twice or using a system of equations.

Key Terms and Formulas

• Integration by Parts Formula:

• Recall derivatives and integrals of and .

Step-by-Step Guidance

1. Let and (or vice versa; both approaches work, but this is common).

2. Compute and .

3. Apply the integration by parts formula to get a new integral involving .

4. Apply integration by parts a second time to the new integral.

5. After the second application, you will have an equation involving the original integral. Set up this equation, but do not solve for the integral yet.

Try solving on your own before revealing the answer!

Q4. Integrate:

Background

Topic: Trigonometric Integrals

This question tests your ability to integrate products of trigonometric functions, often using power-reduction or double-angle identities.

Key Terms and Formulas

• Power-Reduction Identities:

• Product-to-sum identities may also be useful.

Step-by-Step Guidance

1. Rewrite using the power-reduction identities.

2. Multiply out the resulting expression to simplify the integrand.

3. Express the integrand in terms of a sum of cosines (using product-to-sum if needed).

4. Set up the integral in its simplified form, ready to integrate each term separately.

Try solving on your own before revealing the answer!

Q5. Integrate:

Background

Topic: Trigonometric Integrals (Odd Powers)

This question tests your ability to integrate an odd power of sine, which often involves separating one sine factor and using a substitution.

Key Terms and Formulas

• Reduction Formula: For odd powers, write and use .

• Substitution: , .

Step-by-Step Guidance

1. Rewrite as and express in terms of .

2. Use the substitution to rewrite the integral in terms of .

3. Expand the resulting expression and integrate each term with respect to .

4. Convert your answer back to using .

Try solving on your own before revealing the answer!

Q6. Integrate:

Background

Topic: Trigonometric Integrals (Cosecant and Cotangent Powers)

This question tests your ability to integrate products of powers of cosecant and cotangent, often using substitution and reduction formulas.

Key Terms and Formulas

• Reduction Formula: can often be solved by expressing in terms of and derivatives.

• Recall: and .

Step-by-Step Guidance

1. Rewrite as to facilitate substitution.

2. Consider substitution or and compute accordingly.

3. Express the integrand in terms of and .

4. Set up the resulting integral in terms of , ready to integrate term by term.

Try solving on your own before revealing the answer!