The Fundamental Theorem of Calculus states that integration is the inverse process of differentiation. During differentiation, the constant terms of the function being differentiated will disappear. Thus, to account for this in integration, a constant is added to the integral. This constant is usually expressed as c and is called the constant of integration.

The integral of a polynomial ax^n is given by ax^(n + 1)/(n + 1) + c, where c is the constant of integration.

During differentiation, the constant terms of the function being differentiated will disappear. Thus, to account for this in integration, a constant is added to the integral. This constant is usually expressed as c and is called the constant of integration.

Indefinite integrals do not have an evaluated constant of integration. They represent a family of functions which have the same derivative, and their only difference is the value of the constant of integration.

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These equations are the indefinite integrals of the gradient function. So, integrating gives y = x⁴/4 + 5x³/3 − 5x² + 4x + c, where c is the constant of integration. This is the equation form of all functions whose derivative follows the gradient function.

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Differentiating the equation gives dy/dx = 10cosx + 6. Integrating this gives y = 10sinx + 6x + c, where c is the constant of integration. This shows that for c = 10, integrating has undone the differentiation, as stated by the Fundamental Theorem of Calculus.

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By the Fundamental Theorem of Calculus, differentiating the integral will give the function itself, therefore the function is y = cosx + 20. To find the derivative of the function, dy/dx = −sinx.

The integral is given by 5x² + 8x + c, where c is the constant of integration.

5x² + 8x + c

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By the Fundamental Theorem of Calculus, differentiating the integral will give the function itself, therefore the function is y = 36x² + 21. To find the derivative of the function, dy/dx = 72x.