Question

basic integration formulas

Answer 1

The notation, which we're stuck with for historical reasons, is as peculiar as the notation for derivatives: the integral of a function f(x)f(x) with respect to xx is written as
∫f(x)dx
∫f(x)dx
The remark that integration is (almost) an inverse to the operation of differentiation means that if
ddxf(x)=g(x)
ddxf(x)=g(x)
then
∫g(x)dx=f(x)+C
∫g(x)dx=f(x)+C
The extra CC, called the constant of integration, is really necessary, since after all differentiation kills off constants, which is why integration and differentiation are not exactly inverse operations of each other.
Since integration is almost the inverse operation of differentiation, recollection of formulas and processes for differentiation already tells the most important formulas for integration:
∫xndx∫exdx∫1xdx∫sinxdx∫cosxdx∫sec2xdx∫11+x2dx=1n+1xn+1+C=ex+C=lnx+C=−cosx+C=sinx+C=tanx+C=arctanx+C unless n=−1
∫xndx=1n+1xn+1+C unless n=−1 ∫exdx=ex+C∫1xdx=ln⁡x+C∫sin⁡xdx=−cos⁡x+C∫cos⁡xdx=sin⁡x+C∫sec2⁡xdx=tan⁡x+C∫11+x2dx=arctan⁡x+C
And since the derivative of a sum is the sum of the derivatives, the integral of a sum is the sum of the integrals:
∫f(x)+g(x)dx=∫f(x)dx+∫g(x)dx
∫f(x)+g(x)dx=∫f(x)dx+∫g(x)dx
And, likewise, constants ‘go through’ the integral sign:
∫c⋅f(x)dx=c⋅∫f(x)dx