for integration:
Indefinite integral is without limits,
I(x) = ∫ f(x).dx = F(x) +C where F(x) is the integral of f(x), C is arbitrary const.
Definite integral has limits and represents area (positive) under curve between the limits,
A = ∫ [a,b] f(x).dx = F(b) - F(a) this is called The foundamental theorem of calculus.
simply add areas identified where f(x) crosses the x-axis.
Some important integrals are:
∫ a.f(x).dx = a.∫f(x).dx i.e. a simple coef. can be taken outside integrand
I(x^n) = (1/n)x^n+1
I(sin ax) = - (1/a)cos (ax)
I(cos ax)= (1/a)sin (ax)
I(1/(x+a)) = log(x+a)
Use substitutions you can need to practice a lot of these:
e.g. ∫ 1/(x+1)² .dx subs: u = 1/(x+1) => - du = 1/(x+1)² .dx so
I(u) = ∫ -1 .du = - u +c ... notice carefully my notation
I(x) = -1/(x+1) +C
learn and practice your parts integration: ∫ u v'.dx = uv - ∫ u' v.dx - very important!
make sure you study some applied calculus v = ds/dt, a = v. dv/ds = dv/dt
and EOM (equation of motion) type problems,
e.g. the height of a particle h(t) = -16t^2 + vt +s etc. understand initial conditions
at t=0 height = 100 => h(0) = 100
I could go on and on but the above are the main areas where students of calculus get into difficulty.
get a decent book on calculus the ones used by undergraduates.