Question

Choosing a Formula In Exercises $73-76,$ state the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate. $$ \int x \sec \left(x^{2}+1\right) \tan \left(x^{2}+1\right) d x $$

Name: Problem 74, Chapter 8: Calculus of a Single Variable
Uploaded: 2021-03-03T08:06:03-08:00
Duration: 3 min 9 s
Channel: Dwijendra Rao
Description: Choosing a Formula In Exercises 73-76, state the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate. ∫x sec(x^2+1) tan(x^2+1) d x

   Choosing a Formula In Exercises $73-76,$ state the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate. 
$$
\int x \sec \left(x^{2}+1\right) \tan \left(x^{2}+1\right) d x
$$

Calculus of a Single Variable

Ron Larson, Bruce… 10th Edition

Chapter 8, Problem 74

Instant Answer

Solved by Expert Dwijendra Rao

03/03/2021

Step 1

We can rewrite $x \sec(x^2+1) \tan(x^2+1)$ as $\frac{x \sin(x^2+1)}{\cos^2(x^2+1)}$. Show more…

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Choosing a Formula In Exercises $73-76,$ state the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate. $$ \int x \sec \left(x^{2}+1\right) \tan \left(x^{2}+1\right) d x $$

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Key Concepts

U-Substitution

U-substitution is a method to simplify integration by replacing a composite part of the integrand with a single variable. By letting u equal the inner function, the derivative du appears (up to a constant), allowing the integral to transform into a simpler form that is easier to evaluate. This technique essentially reverses the application of the chain rule in differentiation.

Standard Integration Formulas

Standard integration formulas are known antiderivatives of common functions. In many cases, after a u-substitution, the resulting integrand will match a standard form, such as the derivative of a basic trigonometric function. Recognizing these forms helps in quickly identifying the antiderivative without further complicated manipulation.

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