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anonymous

  • 4 years ago

Steps for integrating

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  1. anonymous
    • 4 years ago
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    \[\int\limits_{\pi/4}^{\pi/2} (2\sin \theta)^2 d \theta\]

  2. anonymous
    • 4 years ago
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    i take it this is the same as \[4\int \sin^2(x)dx\] right?

  3. anonymous
    • 4 years ago
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    Yes

  4. dumbcow
    • 4 years ago
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    integration by parts twice

  5. anonymous
    • 4 years ago
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    gimmick is to write \[\sin^2(x)=1-\cos^2(x)=\frac{1}{2}-\frac{1}{2}\cos(2x)\] one of those "double angle formula's backwards"

  6. anonymous
    • 4 years ago
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    best trick is actually to look in the back of the textbook for the "reduction formulas" but if you recall all those annoying trig identities, this is the "lowering powers" formula

  7. anonymous
    • 4 years ago
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    Ah, thanks. I get it.

  8. anonymous
    • 4 years ago
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    yw

  9. amistre64
    • 4 years ago
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    sin reduction:\[\int sin^ndx=-\frac{1}{n}sin^{n-1}cos+\frac{n-1}{n}\int sin^{n-2}dx\]

  10. anonymous
    • 4 years ago
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    yeah that one that i can't remember. the entire content of most calc 2 courses is contained on the back two pages of the text

  11. anonymous
    • 4 years ago
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    I've never seen that formula but wouldn't that give me another answer?

  12. amistre64
    • 4 years ago
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    the answer might "look" different, but since trig has identities that are equal it will have the same value in the end

  13. amistre64
    • 4 years ago
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    using integration by parts ends up with that formula for reduction

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