15A - Introduction to integration
Integration, also known as antidifferentiation, is the reverse process of differentiation. That is, we start with a derivative function and compute the original function \[\int f'(x) dx=f(x)\].
The constant of integration:
When a function is derived, any constant terms are lost in the process. For example:
\[\frac{d}{dx}(x^2+10)=2x\]
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\[\frac{d}{dx}(x^2)=2x\]
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\[\frac{d}{dx}(x^2-6)=2x\]
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Therefore, the antiderivative of \(2x\) is not unique. To account for this, we must include a constant of integration. \[\therefore \int f'(x) dx=f(x)+c\] Returing to our example \(\int 2x dx = x^2+c\), where \(c\in R\).
Indefinite integrals and their properties:
Indefinite integrals have no bounds/terminals of integration. By convention, the antiderivative of \(f\) is given by \(F\). \[\int f(x) dx = F(x)+c\] When working with indefinite integrals, the following properties apply:
- For a function \(f(x)\) and a constant \(k\in R\): \[\int kf(x)dx=k \int f(x) dx\]
- For functions \(f(x)\) and \(g(x)\): \[\int f(x)\pm g(x) dx=\int f(x) dx \pm \int g(x) dx\]
15A - Example 1: Calculating antiderivatives (CAS)
Using a CAS calculator, compute the following:
(a) \(\int 5 dx\) (b) \(\int 3y^2 dy\) (c) \(\int 5t^2+7t^4 dt\) (d) \(\int \frac{1}{x^2}dx\) |
15A - Example 1: Video Example
15A - Example 1: Practice
Question 1: Find the antiderivative of \(3x^2\). Question 2: Find \(\int 3t^5-2t dt\). Question 3: If \(g'(x)=4\sqrt{x}+3\), find \(g(x)\). 15A - Example 1: Solutions
Question 1: \[\int 3x^2 dx=x^3+c\] Question 2: \[\int 3t^5-2t dt=\frac{t^6}{2}-t^2+c\] Question 3: \[g(x)=\int 4\sqrt{x}+3dx=\frac{8x^{\frac{3}{2}}}{3}+3x+c\] |