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Math Help - integration questions stuck :(

  1. #1
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    Question integration questions stuck :(

    Hey guy i gt this questions on integration which i am not sure how to approach. I have uploaded the question via jpg file.

    For the first questions i attempted to intergrate by parts and let u=e^tanx
    du= tane^tanx.dx, dv/dx = cos^2x and v = (sin^3x)/3, am i correct so far?

    As a results i was left with:

    e^tanx . (sin^3x/3) - Int (sin^3X/3) . tane^tanx

    what do i do now if in fact i am on the correct path i would assume to integrate it again but that seems too difficult and deafeats the purpose of the IBP. Was i supposed to use IBP? Any help would be greatly appreciated.
    Attached Thumbnails Attached Thumbnails integration questions stuck :(-integral-question.jpg  
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  2. #2
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    Quote Originally Posted by Latkan View Post
    Hey guy i gt this questions on integration which i am not sure how to approach. I have uploaded the question via jpg file.

    For the first questions i attempted to intergrate by parts and let u=e^tanx
    du= tane^tanx.dx, dv/dx = cos^2x and v = (sin^3x)/3, am i correct so far?

    As a results i was left with:

    e^tanx . (sin^3x/3) - Int (sin^3X/3) . tane^tanx

    what do i do now if in fact i am on the correct path i would assume to integrate it again but that seems too difficult and deafeats the purpose of the IBP. Was i supposed to use IBP? Any help would be greatly appreciated.
    This looks like a simple substituion

    \int^{\pi/4}_{0} e^{\tan x} \sec^2(x)

    Because \sec^2 x = \frac{1}{\cos^2 x}

    Let u = \tan x. Hence du = \sec^2 x\,dx<br />


    Now to work out our limits in terms of u. For this simply sub the limit into our expression for u.

    \text { Lower Limit } = \tan 0 = 0

    \text { Higher Limit } = \tan \left(\frac{\pi}{4}\right) = 1


    Now we can put all these expressions into the integral to give:

    \int^1_0 \, e^u\,du

    Which is a very simple integral and can be solved directly because we changed the limits
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