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P3 Integration watch

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    Integrate with respect to x:

    1) (x+2) / x(x-1)

    2) (sec^2 x) / (1+tanx)^3

    3) sin^2 2x

    4) x^2 / (x-2)

    5) (sinx + 2cosx)^2

    Answers:

    1) 3ln(x-1) - 2ln(x)
    2) - 0.5 (1+tanx)^ -2
    3) 0.5x - 0.125sin4x
    4) 0.5x^2 + 2x + 4 ln(x-2)
    5) 2.5x + 0.75sin2x - cos2x.

    Cheers
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    1) (x+2)/x(x-1) = 3/(x-1)-2/x (partial fractions)
    ∫f(x)dx=3ln|x-1|-2ln|x|+C

    2) ∫sin²2x.dx = ∫(½-½cos4x).dx = ½x-1/8sin4x +C

    3) sec²x/(1+tanx)³
    u=1+tanx => du/dx=sec²x => du=sec²x.dx
    => ∫sec²x/(1+tanx)³dx = ∫u^-3.du = -½u^-2+C
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    Hi,

    Referring to the 5 questions below. I am offering to do these questions for you for free and any other in future for only £25/ year. Mod Edit - Link removed

    1) (x+2) / x(x-1)

    2) (sec^2 x) / (1+tanx)^3

    3) sin^2 2x

    4) x^2 / (x-2)

    5) (sinx + 2cosx)^2

    Ernest
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    2. Use the substitution u=1+tanx with du=sec²x dx.
    3. Use the identity cos4x = 1 - 2sin²2x.
    4. Use long division to split it into proper fractions.
    5. (sinx + 2cosx)^2 = sin²x + 2(2sinxcosx) + 4cos²x = 1 + 2sin2x + 3cos²x. Now use a variant of the identity used in 3 to integrate cos²x.
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    ernmaths, I seriously doubt you will find any customers here, seeing as many members will do what you're offering for free.
 
 
 
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Updated: May 5, 2005

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