High School

Find an antiderivative of the given function [tex]f(x) = 21x^{2} + 12x - 5[/tex].

A. [tex]8x^{3} + 6x^{2} - 5x[/tex]
B. [tex]7x^{3} + 6x^{2} - 5x[/tex]
C. [tex]7x^{3} + 7x^{2} - 5x[/tex]
D. [tex]7x^{3} + 6x^{2}[/tex]

Answer :

The antiderivative of the function f(x) = 21x^(2) + 12x -5 is F(x) = 7x^(3) + 6x^(2) - 5x + C, where C is the constant of integration.

The subject of the question is related to calculus in Mathematics, specifically about finding the antiderivative of a given function. The function given seems a little jumbled, but the first one listed, f(x) = 21x^(2) + 12x - 5, is clear.

To find the antiderivative, we use the power rule for integration which states that the integral of x^n dx is (1/(n+1))*x^(n+1). Applying this rule to each term:

The antiderivative of 21x^(2) is (21/3)*x^(3) = 7x^(3)

The antiderivative of 12x is (12/2)*x^(2) = 6x^(2)

The antiderivative of -5 is -5x

So the antiderivative of the function f(x) = 21x^(2) + 12x - 5 is F(x) = 7x^(3) + 6x^(2) - 5x + C, where C is the constant of integration.

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