Example: Antiderivative |
| Example |
Find the antiderivative of \( \quad f(x)= \displaystyle\frac{3x^3+5x}{x} \)
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see IndInt |
| Solution: |
Step 1, Algebra |
Step 2, Calculus |
more Algebra |
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| more Algebra |
Alg Step 0.1 |
Alg Step 0.2 |
Both Steps Together |
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| Step 1. |
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sWe split the single fraction up into two fractions that have a
common denominator. \[ \displaystyle\frac{3x^3+5x}{x}= \frac{3x^3}{x} + \frac{5x}{x} \]
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| Step 2. |
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sSimplify the two fractions.
\[ \displaystyle\frac{3x^3}{x} + \frac{5x}{x} = 3x^2 + 5 \]
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| Both Steps |
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sThe algebra steps combined.
\[ \displaystyle f(x) = \frac{3x^3+5x}{x} = \frac{3x^3}{x} + \frac{5x}{x} = 3x^2 + 5 \]
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| 1. |
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Use algebra to simplify \( f(x) \) to \[ f(x) = 3x^2 + 5 \]
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| 2. |
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Use the power rule (in reverse), to get following antiderivative
\[F(x) = 3\frac{x^3}{3} + 5x + C = x^3 + 5x + C .\]
Cancel the \(3\)s to get a very friendly antiderivative.
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| N. |
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Note: the power rule of differentiation produces functions that are sums and differences of monomial-like
expressions (when applied to same).
\[ g(x)= a\cdot x^m \pm b\cdot x^n \cdots \]
Our function \( f(x) \) does not look like that (like \( g(x) \) ) so in order to use the
power rule to find antiderivatives, we will first need to use some algebra to make our \( f(x) \)
have the correct form.
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