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P3 integration watch

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    I seem to be stuck on quite a few questions! Please help!

    1) integrate tan(x + Pi/4)

    2) integrate xsec^2x

    3) integrate x^2 e^x^3

    I never know whether to do substitution or integration by parts

    Thank you x
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    1) Substitution of u = cos (x + pi/4)
    2) Parts, u = x, dv/dx = sec^2 x
    3) Substitution u = x^3
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    ok i can do number 3 now..still dont understand 1 and 2 though, could you explain a bit more please? xx
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    1) int tan(x + pi/4) = int sin(x + pi/4)/cos(x + pi/4) Let u = cos(x + pi/4), du/dx = -sin(x + pi/4) so dx = -du/sin(x + pi/4)
    Putting these bits into the integration, we get int sin(x + pi/4)/u * -du/sin(x + pi/4) = int -1/u du = -ln|u| + c = -ln|cos(x + pi/4)| + c = ln|sec(x + pi/4)| + c

    2) int xsec^2 x Let u = x, dv/dx = sec^2 x so du/dx = 1, v = tanx
    int xsec^2 x = xtanx - int tanx dx = xtanx + ln|cosx| + c
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    1)

    ∫tan(x + pi/4)

    = ∫sin(x + pi/4) / cos(x + pi/4)dx

    let u = cos (x + pi/4)

    so du/dx = -sin(x + pi/4)

    so replace cos(x + pi/4) with u and dx by du/(-sin(x + pi/4))

    -∫1/u du

    = -ln(cos(x + pi/4)) + C

    _______________________________

    ∫xsec²x dx

    integration by parts:

    u= x dv= sec²x

    du=dx v= tanx

    by definition the integral is

    uv - ∫v(du)

    = xtanx - ∫tanx dx

    using the answer from q1

    = xtanx + lncosx + C

    Hope this helps.
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    (Original post by jazzyj)
    ok i can do number 3 now..still dont understand 1 and 2 though, could you explain a bit more please? xx
    2) INT x.(sec x)^2 dx

    use parts, making u=x, dv/dx=(secx)^2
    so, du/dx=1 and v=(tan x)
    then just sub into the formula
    • Thread Starter
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    thank you all so much!

    by the way, is it just a standard result that you have learnt that the integral of tanx is ln cosx?
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    (Original post by jazzyj)
    thank you all so much!

    by the way, is it just a standard result that you have learnt that the integral of tanx is ln cosx?
    The integral of tanx is ln|secx| = - ln|cosx| You can show it the same way as Q1 - substitution of u = cosx, du/dx = -sinx etc.
 
 
 
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