601PolynomialExamples

= Antidifferentiating Polynomials =

math . \qquad \displaystyle{ \int } ax^n \, dx = \dfrac {ax^{n+1}}{n+1} + c, \; n \ne -1 \qquad. math

Example 1

math . \qquad \displaystyle{ \int } x^3 + 4x^2 - x + 3 \, dx \qquad. \\ . \\ . \qquad = \dfrac {x^4}{4} + \dfrac {4x^3}{3} - \dfrac{x^2}{2} + 3x + c \qquad. math

Example 2

math \\ . \qquad \displaystyle{\int} \, \dfrac {1}{x^3} + \dfrac {4}{x^2} \, dx = \int {x^{-3} + 4x^{-2} }dx \qquad. \\ .\\ . \qquad \qquad \qquad \quad = \dfrac {x^{-2}}{-2} + \dfrac {4x^{-1}}{-1} + c \qquad. \\ . \\ . \qquad \qquad \qquad \quad = \dfrac {-1}{2x^2} - \dfrac {4}{x} + c math

Example 3

math . \qquad \displaystyle{ \int } \, \sqrt{x} \, dx = \displaystyle{ \int } x^{ \frac{1}{2} } \, dx \qquad. \\.\\ math

math \\ . \qquad \qquad \quad = \dfrac {x^{ \frac{3}{2}}} {\frac{3}{2}} + c \qquad. \\ . \\ . \qquad \qquad \quad = \dfrac {2 \sqrt{x^3}} {3} + c math

Example 4

... ... Find the equation of the curve, //**g(x)**//, ... ... given that **//g(1) = 2//** and

math . \qquad g'(x) = 3 \sqrt{x} + 2 \qquad. math


 * Solution:**

math . \qquad g'(x) = 3 \sqrt{x} + 2 = 3x^{\frac {1}{2}} + 2 \qquad. \\.\\ . \qquad g(x) = \displaystyle{ \int{ 3x^{\frac {1}{2}} + 2 }\, dx } \qquad. \\.\\ . \qquad \qquad = \dfrac {3x^{ \frac{3}{2}} }{ \frac{3}{2} } + 2x + c \qquad. \\ .\\ . \qquad \qquad = \dfrac {2 \times 3x^{ \frac{3}{2}} }{3} + 2x + c \qquad. \\ . \\ . \qquad \qquad = 2x^{ \frac{3}{2}} + 2x + c math

... ... ** We are given the point (1, 2) **

math \\ . \qquad g(1) = 2(1)^{ \frac{3}{2}} + 2(1) + c = 2 \qquad. \\ .\\ . \qquad \Rightarrow \qquad 2 + 2 + c = 2 \qquad. \\ . \\ . \qquad \Rightarrow \qquad c = -2 math

... ... ** Hence **

math . \qquad g(x) = 2x^{ \frac{3}{2}} + 2x - 2 \qquad. math .