Mastering the Integration of Sin(x) dx: Deriving the Equation and Using Trigonometric Identity

integral of sinx dx

To find the integral of sin(x) dx, we can use the integration rules for trigonometric functions

To find the integral of sin(x) dx, we can use the integration rules for trigonometric functions.

The integral of sin(x) dx is equal to -cos(x) + C, where C is the constant of integration.

To see why this is the case, we can differentiate -cos(x) + C and confirm that it gives us sin(x).

Taking the derivative of -cos(x) + C with respect to x, we have:

d/dx (-cos(x) + C) = -(-sin(x)) + 0 = sin(x)

Therefore, the integral of sin(x) dx is indeed -cos(x) + C.

We can also verify this using an alternative method, which involves using the trigonometric identity:

∫ sin(x) dx = ∫ 1 * sin(x) dx

Applying integration by parts, we choose u = sin(x) (so that du = cos(x) dx) and dv = dx (so that v = x):

∫ sin(x) dx = ∫ u dv
= uv – ∫ v du

Substituting the values of u, v, du, and dv, we get:

= x * sin(x) – ∫ x * cos(x) dx

To evaluate the second integral, we can again use integration by parts, this time with u = x (so that du = dx) and dv = cos(x) dx (so that v = sin(x)):

∫ x * cos(x) dx = x * sin(x) – ∫ sin(x) dx

Notice that the second integral on the right side is the same as the original integral we wanted to evaluate.

Rearranging the equation, we have:

2∫ sin(x) dx = x * sin(x)

Dividing both sides by 2, we find:

∫ sin(x) dx = (1/2) * x * sin(x) + C

Since C is a constant, we can replace it with a new constant, resulting in:

∫ sin(x) dx = -cos(x) + C

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

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