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