Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
121. ∫ (1 + x²) / (1 + x³) dx
1
views
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.PE.4
Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ arccos(x / 2) dx
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
8mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Parts
Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals, using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely is crucial to simplify the integral effectively.
Recommended video:
06:18
Integration by Parts for Definite Integrals
Inverse Trigonometric Functions
Inverse trigonometric functions, like arccos(x), are the inverses of trigonometric functions and have specific derivatives. For arccos(x), the derivative is -1/√(1 - x²). Understanding these derivatives helps in differentiating parts of the integrand during integration by parts.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Substitution and Simplification
Substitution involves changing variables to simplify integrals, especially when dealing with composite functions like arccos(x/2). Simplifying the resulting expressions after applying integration by parts often requires algebraic manipulation or trigonometric identities to reach a solvable form.
Recommended video:
04:27Substitution With an Extra Variable
Related Practice
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (sin5t) dt / [1 + (cos5t)²]
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫₋₁⁰ e^x / (e^x + e^(−x)) dx
Textbook Question
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(4x) / (x³ + 4x)] dx
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ dy / (y² − 2y + 2)
Textbook Question
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [t / (t⁴ − t² − 2)] dt