Antiderivative of sinx and cosx: Formula, Proof

The antiderivative of sinx is -cosx+C and the antiderivative of cosx is sinx+C where C denotes a constant. In this post, we will learn what are the antiderivatives of sine functions and cosine functions.

The antiderivative formula of sinx is given by

∫sinx dx = -cosx + C.

The antiderivative formula of cosx is given by

∫cosx dx = sinx + C.

Here C denotes an integral constant.

Antiderivative of sine and cosine functions

Antiderivative of sinx:

The antiderivative of sinx is a function whose derivative will be sinx. We know that

$\dfrac{d}{dx}(- \cos x) = \sin x$.

As the derivative of a constant C is zero, we obtain that

$\dfrac{d}{dx}(- \cos x +C) = \sin x$ …(∗)

As the antiderivative is the opposite process to derivatives, from (∗) we deduce that

The antiderivative of sinx is equal to -cosx +C, where C is a constant.

Antiderivative of cosx:

The antiderivative of cosx is a function whose derivative will be cosx. We know that $\frac{d}{dx}(\sin x) = \cos x$.

⇒ $\dfrac{d}{dx}(\sin x +C) = \cos x$

From the above, it follows that

The antiderivative of cosx is equal to sinx +C, where C is a constant.

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