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. $$ \int t \sin t^{2} d t $$

Name: Problem 11, Chapter 8: Calculus of a Single Variable
Uploaded: 2020-01-18T01:39:29-08:00
Duration: 2 min 6 s
Channel: Pravakar Paul
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. ∫t sint^2 d t

   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. 
$$
\int t \sin t^{2} d t
$$

Calculus of a Single Variable

Ron Larson, Bruce… 10th Edition

Chapter 8, Problem 11

Instant Answer

Solved by Expert Pravakar Paul

01/18/2020

Step 1

Looking at the integrand $t \sin(t^2)$, we can see that it involves the product of a polynomial function ($t$) and a trigonometric function ($\sin(t^2)$). The basic integration formula that can be used for this type of integral is the substitution rule. This 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. $$ \int t \sin t^{2} d t $$

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Key Concepts

u-Substitution

This method involves identifying an inner function (here, u = t²) whose derivative (du = 2t dt) appears in the integrand. In problems like this, recognizing that a part of the integrand is the derivative of another part simplifies the integration process by transforming the integral into one involving u and du, which is easier to handle.

Basic Integration Formula for Trigonometric Functions

Once the substitution is made, the integral often reduces to a standard form that can be directly integrated. In this example, the transformed integral becomes a constant factor times the integral of sin u, a basic formula known to have the antiderivative –cos u + C. This underscores the importance of knowing basic antiderivatives of common trigonometric functions.

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