## ∫cos x

### The integral of cos x is given by

∫ cos x dx = sin x + C

where C is the constant of integration

The integral of cos x is given by

∫ cos x dx = sin x + C

where C is the constant of integration.

To understand why this is the case, we can start by considering the derivative of sin x, which is cos x. Therefore, we can say that the antiderivative of cos x is sin x. However, because integration is the reverse of differentiation, there exists an arbitrary constant C that needs to be added to the antiderivative. This constant accounts for all the possible functions whose derivative is equal to cos x.

In simple terms, the integral of cos x represents the function whose derivative is equal to cos x. Hence, integrating cos x will result in sin x (the antiderivative) plus a constant C.

It’s important to note that the integral of cos x can also be expressed using the trigonometric identity:

∫ cos x dx = sin x + C = -cos(x + π/2) + C

This is because sin x and -cos(x + π/2) are equivalent functions; they differ by a phase shift of π/2.

Ultimately, the answer to the integral of cos x is sin x + C, where C represents all possible constants of integration.

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