Question

Verify the given integration result by differentiation and the Chain Rule. $$ \int x \sin x^{2} d x=-\frac{1}{2} \cos x^{2}+C $$

Name: Problem 37, Chapter 5: Calculus, Early Transcendentals
Uploaded: 2021-08-07T04:29:17-08:00
Duration: 1 min 41 s
Channel: Kian Manafi
Description: Verify the given integration result by differentiation and the Chain Rule. ∫x sinx^2 d x=-(1)/(2) cosx^2+C

   Verify the given integration result by differentiation and the Chain Rule.
$$
\int x \sin x^{2} d x=-\frac{1}{2} \cos x^{2}+C
$$

Calculus, Early Transcendentals

Dennis G. Zill,… 4th Edition

Chapter 5, Problem 37

Instant Answer

Solved by Expert Kian Manafi

08/07/2021

Step 1

The derivative of the left side, $\int x \sin x^{2} dx$, is $x \sin x^{2}$ because the derivative and the integral cancel each other out. Show more…

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Verify the given integration result by differentiation and the Chain Rule. $$ \int x \sin x^{2} d x=-\frac{1}{2} \cos x^{2}+C $$

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Key Concepts

Chain Rule

The Chain Rule is a fundamental concept in calculus used to differentiate composite functions. When verifying an antiderivative by differentiation, the Chain Rule helps in correctly differentiating functions whose arguments are not the variable of differentiation directly but are themselves functions of that variable.

Differentiation for Verification

Differentiation is used to check the correctness of an antiderivative. According to the Fundamental Theorem of Calculus, if you differentiate the result of an integration, you should recover the original integrand. This reversal process is a crucial verification method in integration.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that differentiation is essentially the inverse process of integration. By differentiating an integrated result and obtaining the original integrand, one confirms the accuracy of the integration process.

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