.: integrals :.

     Integration is a core concept of calculus.  Given a function f(x) of a real variable x and an interval [a.b] of the real line, the area is equal to the area of a region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b, with areas below the x-axis being subtracted.

     The term "integral" may also refer to the notion of the antiderivative, a function F whose derivative is the given function f.

basic formulas (S represents the integral symbol):

S x^(n) dx = [ x^(n+1) / (n+1) ] + C
S 1 / x dx = [ ln |x| ] + C
S sin(x) dx = [ -cos(x) ] + C
S cox(x) dx = [ sin(x) ] + C
S e^(x) dx = [ e^(x) ]+ C
S a^(x) dx = [ a^(n) / ln(a) ] + C
S ln(x) dx = [ (x) ln(x) - x ] + C

     Here are two practice problems which demonstrate two of the rules listed above.  To see the answer click the question mark at the end of the problem.

Problem One: S x^(3) dx = ?

Problem Two: S ln(5) dx = ?