## Indefinite Integral

 3.2 Indefinite Integral

• For a constant $$a$$,
 $$\int a \ dx = ax +c$$, where $$a$$ and $$c$$ are constants.

• For a function $$ax^n$$,
 $$\int ax^n \ dx = \dfrac{ax^{n+1}}{n+1}$$, where $$a$$ and $$c$$ are constants, $$n$$ is an integer and $$n \neq -1$$.

• The function $$ax+c$$  and  $$\dfrac{ax^{n+1}}{n+1} +c$$  are known as indefinite integrals for a constant $$a$$ with respect to $$x$$ and function $$ax^n$$ with respect to $$x$$ respectively

Constant of Integration, $$c$$
 The constant of integration, $$c$$ in an indefinite integrals are different and is added as part of indefinite integral for a function such as: $$\int 5 \ dx = 5x +c$$

Remark
 $$\int ax^n \ dx = a \int x^n \ dx$$

Example

Integrate each of the following with respect to $$x$$ :

 (a) $$-0.5$$ (b) $$\int \dfrac{2}{x^2} \ dx$$ Solution: (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}

• If $$f(x)$$ and $$g(x)$$ are functions, then
 $$\int [f(x) \pm g(x)] \ dx = \int f(x) \ dx \pm g(x) \ dx$$

 Example Find the integral for the following $$\int (x-2)(x+6) \ dx$$ Solution: \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}

• Substitution method can be used for function $$(ax+b)^n$$, where $$a$$ and $$b$$ are constants, $$n$$ is an integer and $$n \neq -1$$
 $$\int (ax+b)^n \ dx = \dfrac{(ax+b)^{n+1}}{a(n+1)} +c$$, where $$a$$ and $$b$$ are constants, $$n$$ is an integer and $$n \neq -1$$.

• In general,
 Given the gradien function $$\dfrac{dy}{dx} = f'(x)$$, the equation of curve for the function is $$y = \int f'(x) \ dx$$.

Example

 (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. Solution: (a) Let $$u=5x+2$$, then, \begin{aligned} \dfrac{dy}{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 $$y=5x^3 +2x^2-3x+2$$

## Indefinite Integral

 3.2 Indefinite Integral

• For a constant $$a$$,
 $$\int a \ dx = ax +c$$, where $$a$$ and $$c$$ are constants.

• For a function $$ax^n$$,
 $$\int ax^n \ dx = \dfrac{ax^{n+1}}{n+1}$$, where $$a$$ and $$c$$ are constants, $$n$$ is an integer and $$n \neq -1$$.

• The function $$ax+c$$  and  $$\dfrac{ax^{n+1}}{n+1} +c$$  are known as indefinite integrals for a constant $$a$$ with respect to $$x$$ and function $$ax^n$$ with respect to $$x$$ respectively

Constant of Integration, $$c$$
 The constant of integration, $$c$$ in an indefinite integrals are different and is added as part of indefinite integral for a function such as: $$\int 5 \ dx = 5x +c$$

Remark
 $$\int ax^n \ dx = a \int x^n \ dx$$

Example

Integrate each of the following with respect to $$x$$ :

 (a) $$-0.5$$ (b) $$\int \dfrac{2}{x^2} \ dx$$ Solution: (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}

• If $$f(x)$$ and $$g(x)$$ are functions, then
 $$\int [f(x) \pm g(x)] \ dx = \int f(x) \ dx \pm g(x) \ dx$$

 Example Find the integral for the following $$\int (x-2)(x+6) \ dx$$ Solution: \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}

• Substitution method can be used for function $$(ax+b)^n$$, where $$a$$ and $$b$$ are constants, $$n$$ is an integer and $$n \neq -1$$
 $$\int (ax+b)^n \ dx = \dfrac{(ax+b)^{n+1}}{a(n+1)} +c$$, where $$a$$ and $$b$$ are constants, $$n$$ is an integer and $$n \neq -1$$.

• In general,
 Given the gradien function $$\dfrac{dy}{dx} = f'(x)$$, the equation of curve for the function is $$y = \int f'(x) \ dx$$.

Example

 (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. Solution: (a) Let $$u=5x+2$$, then, \begin{aligned} \dfrac{dy}{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 $$y=5x^3 +2x^2-3x+2$$