For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

Aug 20, 2014 · The integral is also known (less commonly) as the anti-derivative, because integration is the inverse of differentiation (loosely speaking). Integrals are indefinite when there are no bounds imposed, and the result is a family of functions (dependent on the variable of integration) and separated only by an arbitrary additive constant.

Sep 27, 2013 · My HW asks me to integrate $\\sin(x)$, $\\cos(x)$, $\\tan(x)$, but when I get to $\\sec(x)$, I'm stuck.

Jan 20, 2021 · $\begingroup$ "Answers to the question of the integral of 1x are all based on an implicit assumption that the upper and lower limits of the integral are both positive real numbers" --- not completely correct: if they are both negative it also works. This is an improper integral and does not converge in the remaining cases. $\endgroup$ –

The integral which you describe has no closed form which is to say that it cannot be expressed in elementary functions. For example, you can express $\int x^2 \mathrm{d}x$ in elementary functions such as $\frac{x^3}{3} +C$. However, the indefinite integral from $(-\infty,\infty)$ does exist and it is $\sqrt{\pi}$ so explicitly:

Thus, it cannot be integrated in finite elementary symbols, which is why any answer to the integral requires infinitely many symbols, or a new notation. However, I obviously used some heavy results. If you want to see the proof of these results, I recommend researching differential algebra.

Dec 2, 2014 · What is the integral of $\sin(\cos x)$ ? So glad you asked ! :-) Although the indefinite integral does not possess a closed form, its definite counterpart can be expressed in terms of certain special functions, such as Struve H and Bessel J.

For a definite integral with a variable upper limit of integration $\int_a^xf(t)\,dt$, you have ${d\over dx} \int_a^xf(t)\,dt=f(x)$. For an integral of the form $$\tag{1}\int_a^{g(x)} f(t)\,dt,$$ you would find the derivative using the chain rule. As stated above, the basic differentiation rule for integrals is:

Feb 21, 2018 · I was trying to do this integral $$\int \sqrt{1+x^2}dx$$ I saw this question and its' use of hyperbolic functions. I did it with binomial differential method since the given integral is in a form of $\int x^m(a+bx^n)^p\,dx$ and I spent a lot of time on it so I would like to see if it can be done this way and where did I go wrong. $$\int(1+x^2 ...

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.