∫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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