## ∫sin x

### The integral of sin(x) can be calculated using the technique of integration by substitution

The integral of sin(x) can be calculated using the technique of integration by substitution. Let’s go through the steps to find the antiderivative of sin(x):

Step 1: Identify the Inner Function

In this case, the inner function is x.

Step 2: Find the Derivative of the Inner Function

The derivative of x with respect to x is 1.

Step 3: Set Up the Substitution

Let u be the inner function, so u = x.

Step 4: Calculate du

Since u = x, du/dx = 1. Multiply both sides by dx to isolate dx. du = dx.

Step 5: Substitute and Simplify

Substituting x with u in the integral, we have ∫sin(x)dx = ∫sin(u)du.

Step 6: Integrate with Respect to u

The integral of sin(u) with respect to u is -cos(u).

Step 7: Replace u with the Original Variable

Since u = x, we have -cos(u) = -cos(x).

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

In conclusion, ∫sin x dx = -cos(x) + C.

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