Match the antiderivative with the correct integral....

Match the antiderivative with the correct integral. [Integrals are labeled (a), (b), (c), and (d).]
(a) $\int \ln x d x$
(b) $\int x \sin x d x$
(c) $\int x^{2} e^{x} d x$
(d) $\int x^{2} \cos x d x$
$$
y=x^{2} e^{x}-2 x e^{x}+2 e^{x}
$$

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Key Concepts

Verification by Differentiation

Verification by differentiation involves checking the correctness of an antiderivative by differentiating it and confirming that the derivative matches the original integrand. This technique leverages the inverse relationship between differentiation and integration, ensuring that the integrated result is accurate and consistent with the initial function.

Integration by Parts

Integration by parts is a fundamental technique in calculus used for integrating the product of two functions. By choosing one function to differentiate and another to integrate, this method simplifies the integral into a form that is easier to evaluate. It is particularly useful when one of the functions becomes simpler upon differentiation, such as a polynomial, while the other remains manageable upon integration, such as a trigonometric or exponential function.

Antiderivatives and the Fundamental Theorem of Calculus

An antiderivative is a function whose derivative yields the original function, and the process of finding it is known as antidifferentiation or integration. The Fundamental Theorem of Calculus links the concept of the derivative with the integral by stating that integration and differentiation are inverse processes, which allows one to evaluate a definite integral using antiderivatives.

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