Example: Indefinite Integrals |
| Example |
Find the integral \( \quad \displaystyle\int\frac{3x^3+5x}{x}\,dx \)
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See Anti |
| 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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We split the single fraction up into two fractions.
(The two fractions will have a common denominator.)
\[ \displaystyle\frac{3x^3+5x}{x}= \frac{3x^3}{x} + \frac{5x}{x} \]
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What principle is this? |
| Step 2. |
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Simplify the two fractions.
\[ \displaystyle\frac{3x^3}{x} + \frac{5x}{x} = 3x^2 + 5 \]
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What principle is this? |
| Both Steps |
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The 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 \( \quad \displaystyle\int\frac{3x^3+5x}{x}\,dx = \int\, 3x^2+5\, dx \).
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What principle is this? |
| 2. |
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Use the power rule (in reverse), to get the indefinite integral.
\[\int\, 3x^2+5\, dx = 3\frac{x^3}{3} + 5x + C = x^3 + 5x + C .\]
Cancel the \(3\)s to end up with the final, very friendly answer.
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What principle is this? |