15B - Integration of rational powers
As integration is the reverse process of differentiation we reverse the rule for differentiation to antidifferentiate. Therefore, to integrate rational power functions we use the following rule:
\[\int x^n dx=\frac{x^{n+1}}{n+1}+c\]
The rule applies for all \(n\neq -1\). The rule for integrating \(f(x)=x^{-1}\) is covered in Units 3 and 4.
15B - Example 1: Calculating antiderivatives for rational power functions
Evaluate the following integrals:
(a) \(\int -4 dx\) (b) \(\int 3x^2dx\) (c) \(\int 2x^4+2x-6 dx\) |
15B - Example 1: Video Example
15B - Example 1: Practice
Question 1: Find \(\int 3 dx\). Question 2: Find \(\int 2x^3dx\). Question 3: Determine the antiderivative of \(2x-x^4\). 15B - Example 1: Solutions
Question 1: \[\int 3 dx = 3x+c\] Question 2: \[\int 2x^3 dx = \frac{x^4}{2}+c\] Question 3: \[\int 2x-x^4 dx = x^2-\frac{x^5}{5}+c\] |
Finding a unique antiderivative:
When additional information is provided about the antiderivative, it can be used to find a unique antiderivative.
15B - Example 2: Finding a unique antiderivative
Given that \(f'(x)=6x^2+6x-1\).
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15B - Example 2: Video Example
15B - Example 2: Practice
Question 1: Let \(g'(x)=4-2x\). Find the unique solution for \(g(x)\) if \(g(5)=-11\). Question 2: Let \(f'(x)=-3x^2+8x+3\). Find the unique solution for \(f(x)\) if \(f(-1)=7\). 15B - Example 2: Solutions
Question 1: \[\int 4-2xdx=-x^2+4x+c\] \[g(5)=-(5)^2+4(5)+c=-11\] \[\therefore c=-6\] \[\therefore g(x)=-x^2+4x-6\] Question 2: \[\int -3x^2+8x+3dx=-x^3+4x^2+3x+c\] \[\therefore f(-1)=-(-1)^3+4(-1)^2+3(-1)+c=7\] \[\therefore c=5\] \[\therefore f(x)=-x^3+4x^2+3x+5\] |
15B - Example 3: Finding a unique antiderivative
For a curve, \(g\), the gradient of a tangent at any point on the curve is given by \(2x-3\). If the original curve passes through the point \((2,1)\), find the unique rule defining \(g(x)\).
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15B - Example 3: Video Example
15B - Example 3: Practice
Question 1: The derivative of a the function \(h\) is given by the expression \(x^2-1\). Find the unique solution for \(h(x)\) if the graph of \(h\) passes through the point \((3,10)\). Question 2: Let \(\frac{dy}{dx}=\frac{4}{x^2}+3\). Find the rule for \(y\) if the graph passes through the point \((2,5)\) 15B - Example 3: Solutions
Question 1: \[\int x^2-1dx=\frac{x^3}{3}-x+c\] \[\therefore \frac{(3)^3}{3}-3+c=10\] \[\therefore c=4\] \[\therefore h(x)=\frac{x^3}{3}-x+4\] Question 2: \[\int 4x^{-2}+3 dx=-4x^{-1}+3x+c\] \[\therefore \frac{-4}{2}+3(2)+c=5\] \[\therefore c=1\] \[\therefore y=-\frac{4}{x^2}+3x+1\] |