|
|
| |
 |
| |
| Constant of Integration, \(c\) |
|
The constant of integration, \(c\) is added as a part of indefinite integral for a function. For example,
\(\int 5 \, dx=5x+c\).
|
|
| |
| Indefinite Integral for Algebraic Functions |
|
If \(f(x)\) and \(g(x)\) are functions, then
\(\int [f(x)\pm g(x)] \, dx =\int f(x) \, dx \pm \int g(x) \, dx\).
|
|
| |
| Indefinite Integral for Functions in the Form of \((ax+b)^n\) |
|
Substitution method can be use to solve any functions in the form of \((ax+b)^n\). Then,
\(\int (ax+b)^n \, dx=\dfrac{(ax+b)^{n+1}}{a(n+1)}+c\),
where \(a\) and \(b\) are constants, \(n\) is an integer and \(n \neq -1\).
|
|
| |
| Equation of a Curve from its Gradient Function |
| Given the gradient function \(\dfrac{dy}{dx}=f'(x)\), the equation of curve for that function is \(y=\int f'(x) \, dx\). |
|
| |
| Example \(1\) |
|
Integrate each of the following with the respect to \(x\):
(a) \(-0.5\),
(b) \(\int \dfrac{2}{x^2} \, dx\).
|
|
(a)
\(\int -0.5 \ dx = -0.5x +c\).
(b)
\(\begin{aligned} \int \dfrac{2}{x^2} \ dx &= 2 \int x^{-2} \ dx\\\\ &= 2 \begin{pmatrix} \dfrac{x^{-2+1}}{-2+1} \end{pmatrix} +c\\\\ &= -2x^{-1} +c\\\\ &= -\dfrac{2}{x} +c. \end{aligned}\)
|
|
| |
| Example \(2\) |
|
Find the integral for the following
\(\int (x-2)(x+6) \, dx\)
|
|
\(\begin{aligned} \int (x-2)(x+6) \ dx &= \int (x^2 +4x - 12) \ dx\\\\ &= \int x^2 \ dx + \int 4x \ dx-\int12 \ dx\\\\ &= \dfrac{x^3}{3} + \dfrac{4x^2}{2} - 12x +c\\\\ &= \dfrac{x^3}{3} + 2x^2- 12x +c. \end{aligned}\)
|
|
| |
| Example \(3\) |
|
(a) By using the substitution method, find the indefinite integral for the following
\(\int \sqrt{5x+2} \, dx\)
(b) The gradient function of a curve at point \((x,y)\) is given by
\(\dfrac{dy}{dx} = 15x^2 + 4x- 3\)
If the curve passes through the point \((-1, 2)\), find the equation of the curve.
|
|
(a)
Let \(u=5x+2\), so,
\(\begin{aligned} \dfrac{du}{dx} &=5\\\\ dx&= \dfrac{du}{5} \end{aligned}\)
\(\begin{aligned} \int \sqrt{5x+2} \ dx &= \int \dfrac{\sqrt u}{5} \ du\\\\ &= \int \dfrac{u^{\frac{1}{2}}}{5} \ du\\\\ &=\dfrac{2}{15}u^{\frac{3}{2}} +c\\\\ &= \dfrac{2}{15}(5x+2)^{\frac{3}{2}}+c. \end{aligned}\)
(b)
Given \(\dfrac{dy}{dx} = 15x^2 + 4x- 3\).
Then,
\(\begin{aligned} y&=\int (15x^2 +4x-3) \ dx\\\\ y&=5x^3 +2x^2-3x+c. \end{aligned}\)
When \(x=-1\) and \(y=2\),
\(\begin{aligned} 2&=5(-1)^3+2(-1)^2-3(-1)+c\\ c&=2 .\end{aligned}\)
Thus, the equation of the curve is:
\(y=5x^3 +2x^2-3x+2\).
|
|
| |
