Mastering the Integration of Sin x: Two Methods for Finding Its Antiderivative

∫sin x dx

-cos x + c

To integrate sin x dx, we can use integration by parts or the substitution method.

Method 1: Integration by Parts

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

Using the formula for integration by parts, we have:

∫sin x dx = -cos x + ∫cos x dx

We can integrate cos x using the same method. Let u = cos x and dv = dx. Then, du/dx = -sin x and v = x.

Continuing with integration by parts, we have:

∫sin x dx = -cos x + sin x + C, where C is the constant of integration.

Therefore, the antiderivative of sin x is -cos x + sin x + C.

Method 2: Substitution

We can also integrate sin x using substitution. Let u = cos x, then du/dx = -sin x and dx = du/-sin x.

Substituting in the integral, we have:

∫sin x dx = ∫-du/u

Integrating -du/u gives us:

– ln|u| + C

Substituting back u = cos x, we have:

– ln|cos x| + C

Therefore, the antiderivative of sin x is – ln|cos x| + C.

Both methods give us the antiderivative of sin x in different forms.

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