∫sin(x)dx
To solve the integral of sin(x)dx, we can use the mathematical technique of integration by substitution
To solve the integral of sin(x)dx, we can use the mathematical technique of integration by substitution.
Step 1: Let u = cos(x)
Taking the derivative of u with respect to x, we get du/dx = -sin(x)
Rearranging the equation, we have sin(x) = -du/dx
Step 2: Substitute the value of sin(x) and dx in terms of u.
∫sin(x)dx = -∫(-du/dx)dx
Step 3: Simplify the integral using the substitution.
Since -∫(-du/dx)dx can be written as ∫du, the integral simplifies to:
∫sin(x)dx = -∫(-du/dx)dx = ∫du
Step 4: Integrate with respect to u.
The integral of du is simply u + C, where C is the constant of integration.
Therefore, the solution to ∫sin(x)dx is u + C. Substituting the value of u back in terms of x, we get:
∫sin(x)dx = cos(x) + C
In conclusion, the antiderivative of sin(x) with respect to x is cos(x) + C, where C is the constant of integration.
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