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How to Integrate the Sine Function

An integral and derivative are like a mother and a trouble-making two-year-old: one undoes with the other one does. This is why another term for "integral" is "antiderivative." If the derivative of the natural log (ln) of x + 3 is 1/x, for example, the integral of 1/x is the natural log (ln) of x + C. The C represents a constant. It is telling you that you can replace C with any number and get a different function that will have the same derivative.

Things You'll Need

Pencil
Paper

Instructions

- 1
  Write the sine function you want to integrate. For example, write "sin (7x) dx."
- 2
  Identify the coefficient of the x-term in parentheses in your function. The coefficient of a variable like x is any number next to it on the left. For example, 7 is the coefficient of 7x in your function.
- 3
  Write a negative sign followed by the inverse of the coefficient you have identified. The inverse of a number is 1 divided by that number. For example, write "-1/7."
- 4
  Write the term for cosine, which is "cos." For example, write "-1/7cos."
- 5
  Write the same term in parentheses that was in parentheses in the function you are integrating. The term 7x was the term in parentheses in the function you first wrote. For example, write "-1/7cos (7x)."
- 6
  Write a plus sign (+) and a capital C. For example, write "-1/7cos (7x) + C." The integral of the sine function sin (7x) dx is -1/7cos (7x) + C.

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