Textbook Question
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of Ζ and the π-axis. Evaluate the following integrals.
β«βα΅ Ζ(π) dπ
Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.5.40
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« (sinβ΅ π + 3 sinΒ³ πβ sin π) cos π dπ
Verified step by step guidance1
Step 1: Recognize that the integral involves powers of sin(π) multiplied by cos(π). This suggests using a substitution method where u = sin(π).
Step 2: Compute the derivative of u with respect to π: du/dπ = cos(π), which implies that du = cos(π) dπ. Substitute u = sin(π) and du = cos(π) dπ into the integral.
Step 3: Rewrite the integral in terms of u: β« (uβ΅ + 3uΒ³ - u) du. This simplifies the integral into a polynomial form.
Step 4: Apply the power rule for integration to each term of the polynomial: β« uβ΅ du = (uβΆ)/6, β« 3uΒ³ du = (3uβ΄)/4, and β« -u du = -(uΒ²)/2.
Step 5: Combine the results to express the indefinite integral in terms of u, then substitute back u = sin(π) to return to the original variable. Finally, check your work by differentiating the result to ensure it matches the original integrand.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Indefinite Integrals
Indefinite integrals represent a family of functions whose derivative gives the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation, and it is essential for solving problems in calculus involving area under curves and accumulation of quantities.
Recommended video:
05:04
Introduction to Indefinite Integrals
Change of Variables
Change of variables, or substitution, is a technique used in integration to simplify the integrand. By substituting a new variable for a function of the original variable, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with complex functions or when the integrand can be expressed in terms of a simpler function.
Recommended video:
06:35Changing Geometries
Differentiation Check
Checking work by differentiation involves taking the derivative of the result obtained from an indefinite integral to verify its correctness. This process ensures that the original integrand is recovered, confirming that the integration was performed accurately. It serves as a crucial step in validating the solution and reinforcing the relationship between differentiation and integration.
Recommended video:
05:02Determining Differentiability Graphically
Related Practice
Textbook Question
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«ββΒΉ (πβ1) (πΒ²β2π)β· dπ
Textbook Question
Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.
β«α΅ββ Ζ(g(π)) dπ
Textbook Question
Average distance on a triangle Consider the right triangle with vertices (0,0) ,(0,b) , and (a,0) , where a > 0 and b > 0. Show that the average vertical distance from points on the π-axis to the hypotenuse is b/2 , for all a > 0 .
Textbook Question
Derivatives of integrals Simplify the following expressions.
d/dπ β«βΛ£ (β1 + tΒ²) dt (Hint: β«Λ£ββ (β1 + tΒ²) dt = β«β°ββ (β1 + tΒ²) dt + β«Λ£ββ (β1 + tΒ²) dt ) .
1
views
Textbook Question
Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.
β«α΅ββ Ζ(p(π)) dπ