Q1. Integrate by parts: \( \int 4t^2 e^{-t} \, dt \)

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:

• u: a function you choose to differentiate

• dv: a function you choose to integrate

Step-by-Step Guidance

1. Identify parts: Let and . (You could also factor out the 4 and let for simplicity.)

2. Compute and :

   • derivative of with respect to

   • integral of

3. Apply the integration by parts formula: .

4. After applying the formula, you will have a new integral to solve. Repeat integration by parts if necessary.

Try solving on your own before revealing the answer!

Q2. Integrate by parts: \( \int \cos^{-1}(5x) \, dx \)

Background

Topic: Integration by Parts

This question requires you to integrate the inverse cosine function, which is not a standard integral and thus is a good candidate for integration by parts.

Key Terms and Formulas

Integration by Parts Formula:

• Recall that

Step-by-Step Guidance

1. Let and .

2. Compute (using the chain rule) and (integral of ).

3. Apply the integration by parts formula: .

4. Simplify the resulting integral, which will involve .

Try solving on your own before revealing the answer!

Q3. Integrate by parts: \( \int \sin(\ln 4x) \, dx \)

Background

Topic: Integration by Parts (with substitution)

This integral involves a composition of functions, suggesting substitution and possibly integration by parts.

Key Terms and Formulas

Integration by Parts Formula:

Recall that .

Step-by-Step Guidance

1. Consider substitution: Let so that .

2. Express in terms of and .

3. Rewrite the integral in terms of and .

4. Apply integration by parts, choosing appropriate and for the new integral.

Try solving on your own before revealing the answer!

Q4. Integrate: \( \int \sec^{15} x \tan^5 x \, dx \)

Background

Topic: Trigonometric Integrals

This question tests your ability to integrate products of powers of secant and tangent functions, often using substitution or reduction formulas.

Key Terms and Formulas

• Recall:

• Reduction formulas for powers of secant and tangent may be useful.

Step-by-Step Guidance

1. Consider expressing in terms of and .

2. Use substitution: Let , so .

3. Rewrite the integral in terms of and powers of and .

4. Apply the reduction formula or integrate term by term as appropriate.

Try solving on your own before revealing the answer!

Q5. Integrate: \( \int \cos^2 x \sin x \, dx \)

Background

Topic: Trigonometric Integrals

This integral involves powers of sine and cosine, which can often be solved using substitution.

Key Terms and Formulas

• Let , then

Step-by-Step Guidance

1. Let so that .

2. Rewrite the integral in terms of and .

3. Integrate with respect to .

4. Substitute back in terms of after integrating.

Try solving on your own before revealing the answer!

Q6. Integrate: \( \int \csc^5 x \cot^5 x \, dx \)

Background

Topic: Trigonometric Integrals

This integral involves higher powers of cosecant and cotangent, which often require reduction formulas or substitution.

Key Terms and Formulas

• Recall:

• Reduction formulas for powers of cosecant and cotangent may be useful.

Step-by-Step Guidance

1. Express in terms of and .

2. Use substitution: Let , so .

3. Rewrite the integral in terms of and powers of and .

4. Apply the reduction formula or integrate term by term as appropriate.

Try solving on your own before revealing the answer!

Q7. Integrate: \( \int \sqrt{7 - 36x^2} \, dx \)

Background

Topic: Trigonometric Substitution

This integral involves a square root of a quadratic expression, which is a classic case for trigonometric substitution.

Key Terms and Formulas

• For , use

• Recall:

Step-by-Step Guidance

1. Let , so .

2. Substitute and into the integral and simplify the square root.

3. Integrate with respect to using standard trigonometric integrals.

4. Convert back to using the original substitution.

Try solving on your own before revealing the answer!

Q8. Integrate: \( \int x^3 \sqrt{1 + 36x^2} \, dx \)

Background

Topic: Trigonometric Substitution / Substitution

This integral involves a polynomial times a square root, suggesting substitution or trigonometric substitution.

Key Terms and Formulas

• Let , then

• Alternatively, use for trigonometric substitution.

Step-by-Step Guidance

1. Let , so .

2. Express in terms of and .

3. Rewrite the integral in terms of and integrate.

4. Substitute back in terms of after integrating.

Try solving on your own before revealing the answer!

Q9. Integrate: \( \int \frac{x^3 + 3x}{x + 1} dx \)

Background

Topic: Integration of Rational Functions (Polynomial Division)

This integral involves a rational function where the degree of the numerator is higher than the denominator, so polynomial long division is needed before integrating.

Key Terms and Formulas

• Polynomial long division: Divide by .

• Integrate the resulting polynomial and any remainder over .

Step-by-Step Guidance

1. Perform polynomial long division to write as a polynomial plus a remainder over .

2. Integrate each term of the resulting expression separately.

3. For the term with , recall .

4. Combine all terms and add the constant of integration.

Try solving on your own before revealing the answer!

Q10. Integrate: \( \int \frac{5x - 5}{(x + 3)(x^2 + 1)} dx \)

Background

Topic: Integration of Rational Functions (Partial Fractions)

This integral involves a rational function with a quadratic and a linear factor in the denominator, suggesting partial fraction decomposition.

Key Terms and Formulas

• Partial fraction decomposition: Express as .

• Integrate each term separately.

Step-by-Step Guidance

1. Set up the partial fraction decomposition and solve for , , and .

2. Write the integral as a sum of simpler fractions.

3. Integrate each term: , , and .

4. Combine the results and add the constant of integration.

Try solving on your own before revealing the answer!

Q11. Integrate: \( \int \frac{16x + 60}{x^2 + 8x + 15} dx \)

Background

Topic: Integration of Rational Functions (Partial Fractions)

This integral involves a rational function with a quadratic denominator that can be factored, suggesting partial fraction decomposition.

Key Terms and Formulas

• Factor .

• Set up partial fractions: .

• Integrate each term separately.

Step-by-Step Guidance

1. Factor the denominator and set up the partial fraction decomposition.

2. Solve for and by equating coefficients.

3. Write the integral as a sum of two logarithmic integrals.

4. Integrate each term and combine the results.

Try solving on your own before revealing the answer!

Q12. Integrate: \( \int 5x^2 \sin x \, dx \)

Background

Topic: Integration by Parts

This integral involves a polynomial times a trigonometric function, which is a classic case for repeated integration by parts.

Key Terms and Formulas

Integration by Parts Formula:

Step-by-Step Guidance

1. Let and .

2. Compute and .

3. Apply the integration by parts formula.

4. The resulting integral will again require integration by parts. Repeat as needed.

Try solving on your own before revealing the answer!

Q13. Integrate: \( \int \cos^8 x \sin^5 x \, dx \)

Background

Topic: Trigonometric Integrals

This integral involves powers of sine and cosine, which can be solved using substitution and reduction formulas.

Key Terms and Formulas

• Use substitution: Let , .

• Reduction formulas for powers of sine and cosine.

Step-by-Step Guidance

1. Rewrite as .

2. Let , so .

3. Express the integral in terms of and .

4. Expand and integrate each term in .

Try solving on your own before revealing the answer!

Q14. Use Simpson's Rule with to approximate (round to 6 decimal places)

Background

Topic: Numerical Integration (Simpson's Rule)

This question tests your ability to use Simpson's Rule to approximate a definite integral numerically.

Key Terms and Formulas

Simpson's Rule:

• must be even;

• for

Step-by-Step Guidance

1. Calculate .

2. List all values from to in increments of .

3. Compute for each .

4. Apply Simpson's Rule formula, using the correct coefficients for each .

Try solving on your own before revealing the answer!

Q15. Use the Trapezoidal Rule with to approximate (round to 6 decimal places)

Background

Topic: Numerical Integration (Trapezoidal Rule)

This question tests your ability to use the Trapezoidal Rule to approximate a definite integral numerically.

Key Terms and Formulas

Trapezoidal Rule:

• for

Step-by-Step Guidance

1. Calculate .

2. List all values from to in increments of .

3. Compute for each .

4. Apply the Trapezoidal Rule formula, using the correct coefficients for each .

Try solving on your own before revealing the answer!