Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

8. Definite Integrals

Introduction to Definite Integrals

Calculus

8. Definite Integrals

Introduction to Definite Integrals

Previous problemNext problem

2:56 minutes

Problem 5.R.48

Textbook Question

Evaluating integrals Evaluate the following integrals.

∫₁⁴ ((√v + v) / v ) dv

Verified step by step guidance

Step 1: Simplify the integrand. The given integral is ∫₁⁴ ((√v + v) / v) dv. Break the fraction into two terms: (√v / v) + (v / v). Simplify each term to get (1/√v) + 1.

Step 2: Rewrite the integrand in terms of exponents. Recall that √v = v^(1/2), so 1/√v = v^(-1/2). The integrand becomes v^(-1/2) + 1.

Step 3: Apply the power rule for integration. The integral of v^n is (v^(n+1))/(n+1), provided n ≠ -1. Integrate each term separately: ∫v^(-1/2) dv and ∫1 dv.

Step 4: Compute the antiderivatives. For v^(-1/2), the antiderivative is (v^(1/2))/(1/2) = 2√v. For 1, the antiderivative is v. Combine these results to get the antiderivative: 2√v + v.

Step 5: Evaluate the definite integral. Substitute the limits of integration (1 and 4) into the antiderivative, 2√v + v, and compute the difference between the values at the upper and lower limits.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration

Integration is a fundamental concept in calculus that involves finding the accumulated area under a curve represented by a function. It is the reverse process of differentiation and can be used to calculate quantities such as areas, volumes, and total accumulated change. The integral symbol (∫) denotes the operation, and definite integrals have specified limits, indicating the interval over which the integration is performed.

Simplifying Expressions

Simplifying expressions is a crucial step in calculus that involves rewriting mathematical expressions in a more manageable form. In the context of integrals, this often includes factoring, combining like terms, or reducing fractions to make the integration process easier. For the given integral, simplifying the expression ((√v + v) / v) can help in breaking it down into simpler components that are easier to integrate.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with integration, stating that if a function is continuous on an interval, then the integral of its derivative over that interval equals the change in the function's values at the endpoints. This theorem provides a method for evaluating definite integrals by finding an antiderivative of the integrand, which is essential for solving the integral in the question.

Watch next

Master Definition of the Definite Integral with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.

∫₁³ ƒ(𝓍)/g(𝓍) d𝓍

views