Solving The Integral Of Cos X Using Substitution And Integration By Parts Method

∫cos x

sin x

The integral of cos x can be solved using substitution or integration by parts method.

Method 1: Substitution

Let u = sin x
Then substitute, du = cos x dx
Therefore, the integral becomes:

∫ cos x dx = ∫ du/u (substituting cos x = du)

Integrating ∫ du/u gives us log |u| or log |sin x| (since u = sin x)

Therefore,

∫cos x dx = log |sin x| + C

where C is the constant of integration.

Method 2: Integration by Parts

We can use integration by parts method to solve ∫cos x dx as follows:

Let u = cos x, dv = dx
Then, du = -sin x dx and v = x

Therefore, the integral becomes:

∫cos x dx = uv – ∫v du
Plugging in the values, we get:

∫ cos x dx = x cos x – ∫x (-sin x) dx

This simplifies to:

∫ cos x dx = x cos x + ∫x sin x dx

Now, we can use integration by parts again:

Let u = x, dv = sin x dx
Then, du = dx and v = -cos x

Therefore, the integral becomes:

∫ cos x dx = x cos x + (-x cos x) – ∫-cos x dx

Simplifying further,

∫cos x dx = sin x + C

where C is the constant of integration.

So, both the substitution and integration by parts methods give us the same answer.

Therefore, the integral of cos x is sin x + C (where C is a constant of integration).

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