Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.9.31
Chapter 4, Problem 4.9.31

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (3x + 1) (4 - x) dx

Verified step by step guidance
1
Step 1: Expand the integrand (3x + 1)(4 - x) using the distributive property. Multiply each term in the first parenthesis by each term in the second parenthesis to rewrite the integrand.
Step 2: Combine like terms after expanding. This will simplify the integrand into a polynomial expression.
Step 3: Apply the power rule for integration to each term of the polynomial. Recall that the integral of x^n is (x^(n+1))/(n+1) and the integral of a constant is the constant multiplied by x.
Step 4: Add the constant of integration, C, to the result since this is an indefinite integral.
Step 5: Check your work by differentiating the result. Use the derivative rules to ensure that the derivative of your solution 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:
2m
Was 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 is 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, where we seek a function F(x) such that F'(x) equals the integrand.
Recommended video:
05:04
Introduction to Indefinite Integrals

Integration by Expansion

Integration by expansion involves simplifying the integrand before integrating. This can be done by distributing or expanding the terms within the integral. For example, in the integral ∫ (3x + 1)(4 - x) dx, we first expand the product to obtain a polynomial, which can then be integrated term by term.
Recommended video:
06:18
Integration by Parts for Definite Integrals

Checking Work by Differentiation

After finding an indefinite integral, it is essential to verify the result by differentiation. This involves taking the derivative of the antiderivative obtained and ensuring it matches the original integrand. This step confirms the correctness of the integration process and helps identify any potential errors in calculations.
Recommended video:
06:22
Introduction To Work
Related Practice
Textbook Question

Trajectory high point A stone is launched vertically upward from a cliff 192 ft above the ground at a speed of 64 ft/s. Its height above the ground t seconds after the launch is given by s = -16t² + 64t + 192, for 0 ≤ t ≤ 6. When does the stone reach its maximum height?

Textbook Question

Linear approximation Find the linear approximation to the following functions at the given point a.


g(t) = √(2t + 9); a = -4

Textbook Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = -12x⁵ + 75x⁴ - 80x³

1
views
Textbook Question

Sketch the graph of a continuous function ƒ on [0, 4] satisfying the given properties.


ƒ' (x) = 0 for x = 1 and 2; ƒ has an absolute maximum at x = 4; ƒ has an absolute minimum at x= 0; and ƒ has a local minimum at x = 2.

Textbook Question

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ 0 (eˣ - 1) / (x² + 3x)

Textbook Question

{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.


f(x) = tan x - 2x; x₀ = 1.2