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You'll want to understand the different techniques for different types of integrals. For instance, simple polynomials can be integrated very easily, whereas a product of functions has a special technique called "Integration by Parts" that is used to solve the integral. It simply depends on the format of the integrand (what is inside the integral).
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Is integrals an antonym for apply?
No
Do all functions have integrals?
No, all functions are not Riemann integrable
What is the integral of sinx2?
If you mean the the integral of sin(x2)dx, It can only be represented as an infinite series or a unique set of calculus functions known as the Fresnel Integrals (Pronounced Frenel). These functions, S(x) and C(x) are the integrals of sin(x2) ans cos(x2) respectively. These two integrals have some interesting properties. To find out more, go to: http://en.wikipedia.org/wiki/Fresnel_integral I hope this answers your question.
What comes to mind when you hear the word calculus?
The first thing that come up into my mind is numbers, calculation, integrals and derivatives
Applying calculus to walking the dog?
You could look at the length of the walk and use integrals to determine that.
