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What is the integration of sinehx?
It is cosh(x) + c where c is a constant of integration.
What is integration of 34?
Assuming integration is with respect to a variable, x, the answer is 34x + c where c is the constant of integration.
Why do you need initial condition to solve differential equation?
The solution to a differential equation requires integration. With any integration, there is a constant of integration. This constant can only be found by using additional conditions: initial or boundary.
Why does an answer to an integration problem involve a Constant of Integration?
The indefinite integral is the anti-derivative - so the question is, "What function has this given function as a derivative". And if you add a constant to a function, the derivative of the function doesn't change. Thus, for example, if the derivative is y' = 2x, the original function might be y = x squared. However, any function of the form y = x squared + c (for any constant c) also has the SAME derivative (2x in this case). Therefore, to completely specify all possible solutions, this constant should be added.
Why does you put plus c after integration?
When you find an indefinite integral of a function (ie, the integral of a function without integration limits) you are actually finding the antiderivative of that function. In other words, you are finding the function whose derivative is the function 'inside' the integral sign. Recall that the derivative of a constant is zero. The point here is that you add the 'c' to acknowledge the fact that when the derivative of the result of your integration effort is taken to get the original function it could, or would, have been followed by some unknown constant value that disappeared upon differentiation. That constant is denoted by the 'c'.
