Question

Choosing a Formula In Exercises $5-14$ , select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate. $$ \int(5 x-3)^{4} d x $$

Name: Problem 5, Chapter 8: Calculus of a Single Variable
Uploaded: 2020-01-05T23:44:39-08:00
Duration: 2 min 42 s
Channel: Harriet O'Brien
Description: Choosing a Formula In Exercises 5-14 , select the basic integration formula you can use to find the integral, and identify u and a when appropriate. ∫(5 x-3)^4 d x

   Choosing a Formula In Exercises $5-14$ , select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate. 
$$
\int(5 x-3)^{4} d x
$$

Calculus of a Single Variable

Ron Larson, Bruce… 10th Edition

Chapter 8, Problem 5

Instant Answer

Solved by Expert Harriet O'Brien

01/05/2020

Step 1

The integral given is of the form $\int (ax + b)^n \, dx$, which suggests using the power rule for integration. Show more…

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Choosing a Formula In Exercises $5-14$ , select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate. $$ \int(5 x-3)^{4} d x $$

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Key Concepts

u-Substitution

This concept involves making a substitution (u = g(x)) in an integral to simplify its form. For integrals that include a composite function—especially when one part of the integrand is the derivative of another—setting u equal to the inner function can make the integration process more straightforward. In the context of the given problem, recognizing that u can be set to a linear function is key to applying the substitution method.

Integration Formula for Power Functions

This formula deals with integrating functions of the form (ax + b)^n, where n is a constant. Typically, once the substitution u = ax + b is made, the integral becomes one of the form ?u^n du, which can be directly integrated using the power rule by increasing the exponent by one and dividing by the new exponent. The constant a from the substitution is handled appropriately, often resulting in a factor of 1/a in the antiderivative.

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