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Math Help - Integral - Multiplying by a form of one

  1. #1
    Member RedBarchetta's Avatar
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    Integral - Multiplying by a form of one

    Apparently, I can multiply this by a fraction equal to one that contains some trig functions.

    <br />
\int {\frac{1}<br />
{{1 + \sin x}}dx} <br />

    I can't seem to find this "trick" multiplier simplify this. I've tried cosx/cosx , cscx/cscx, cotx/cotx, etc.
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    Quote Originally Posted by RedBarchetta View Post
    Apparently, I can multiply this by a fraction equal to one that contains some trig functions.

    <br />
\int {\frac{1}<br />
{{1 + \sin x}}dx} <br />

    I can't seem to find this "trick" multiplier simplify this. I've tried cosx/cosx , cscx/cscx, cotx/cotx, etc.
    \frac{1}{1+\sin x} \cdot \frac{1-\sin x}{1-\sin x} = \frac{1-\sin x}{\cos^2 x} = \sec^2 x - \frac{\sin x}{\cos^2 x}
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    Member RedBarchetta's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    \frac{1}{1+\sin x} \cdot \frac{1-\sin x}{1-\sin x} = \frac{1-\sin x}{\cos^2 x} = \sec^2 x - \frac{\sin x}{\cos^2 x}
    Thanks. Is there any way to learn how to recognize these patterns better or a certain technique that you use?
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    Quote Originally Posted by RedBarchetta View Post
    Thanks. Is there any way to learn how to recognize these patterns better or a certain technique that you use?
    A lot of comes from experience.
    The more problems you do the more ideas you get to how to solve problems.
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    Moo
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    Here, you have 1+sin(x)

    Try to find identities with 1 and sin(x). The first that comes in mind is cosē(x)+sinē(x)=1 ---> 1-sinē(x)=cosē(x)=(1-sin(x))(1+sin(x))

    When you're dealing with cot, or tan, it can be interesting to first multiply by cos or sin, regarding the situation.

    When you're dealing with a sum of cos or a sum of sin, recall the identities of addition sin(a+b), cos(a+b)... cos(a)+cos(b)=.... sin(a)+sin(b)=....

    But it's true, like TPH says, a lot comes with experience.
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