The integral of 0 is C, because the derivative of C is zero. Also, it makes sense logically if you recall the fact that the derivative of the function is the function's slope, because any function f (x)=C will have a slope of zero at point on the function.
Answers to the question of the integral of $\frac {1} {x}$ are all based on an implicit assumption that the upper and lower limits of the integral are both positive real numbers.
@user599310, I am going to attempt some pseudo math to show it: $$ I^2 = \int e^-x^2 dx \times \int e^-x^2 dx = Area \times Area = Area^2$$ We can replace one x, with a dummy variable, move the dummy copy into the first integral to get a double integral. $$ I^2 = \int \int e^ {-x^2-y^2} dA $$ In context, the integrand a function that returns ...
The noun phrase "improper integral" written as $$ \int_a^\infty f (x) \, dx $$ is well defined. If the appropriate limit exists, we attach the property "convergent" to that expression and use the same expression for the limit.
A different approach, building up from first principles, without using cos or sin to get the identity, $$\arcsin (x) = \int\frac1 {\sqrt {1-x^2}}dx$$ where the integrals is from 0 to z. With the integration by parts given in previous answers, this gives the result. The distance around a unit circle traveled from the y axis for a distance on the x axis = $\arcsin (x)$. $$\arcsin (x) = \int\frac ...
A different type of integral, if you want to call it an integral, is a "path integral". These are actually defined by a "normal" integral (such as a Riemann integral), but path integrals do not seek to find the area under a curve. I think of them as finding a weighted, total displacement along a curve.
Wolfram Mathworld says that an indefinite integral is "also called an antiderivative". This MIT page says, "The more common name for the antiderivative is the indefinite integral." One is free to define terms as you like, but it looks like at least some (and possibly most) credible sources define them to be exactly the same thing.
Definite integrals of that function are found by numerical methods rather than by finding a closed-form antiderivative. In exercises of this kind usually one gets the value of the integral either from software or from a table in the back of the book.
A scalar line integral is just like a regular integral that they teach to first year Calc students, except the function to be integrated doesn't necessarily have just one variable, and the region being integrated is not restricted to the x axis, but is a one dimensional path on the xy-plane (or whatever, depending on how many variables).
