Integration
Integration is the reversal of differentiation An integral can either be indefinite while it has no numerical value or may definite while have specific numerical values Integration is represented by the sign ?f(x)dx.
Rules of integration
i.The integral of a constant
?adx = ax +c; whereas; a = constant
Illustration
Find the following
a) ?23dx
b) ??2dx ; whereas ? is a variable independent of x, hence it is treated as a constant.
Solution
i. ?23dx = 23x + c
ii. ??2dx. = ?2 x + c
ii. The integral of x raised to the power n
?x^{n} dx = 1/((n+1). x^{n+1}+ c
Illustration
Find the following integrals
a) ?x^{2}dx
b) ?x^{-5/2} dx
Solution
i. ?x^{2}dx = (1/3)x^{3} + c
ii. ?x^{-5/2} dx = -(2/3)x^{3/2} + c
iii). Integral of a constant times a function
? af(x) dx = a ? f(x) dx
Illustration
Determine the given integrals:
i.?ax^{3}dx
ii.?20x^{5}dx
Solution
a). ?ax^{3}dx = a ? x^{3} dx
= (a/4) x^{4} + c
b). ?20x^{5}dx = 20 ?20x^{5}dx
= (10/3) x^{6} +c
iv). Integral of sum of two or more functions
?{f(x) + g(x)} dx = ?f(x)dx + ?g(x) dx
?{f(x) + g(x) + h(x)}dx = ?f(x)dx + ?g(x)dx + ?h(x)dx
Illustration
Find the given
i. ?(4x^{2} + ½ x^{-3}) dx
ii. ?(x^{3/4 }+ 3/7 x^{- ½ }+ x^{5})
Solution
v. Integral of a difference
?{f(x) - g(x)} dx = ?f(x)dx - ?g(x) dx