Problem 8
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![{\displaystyle \displaystyle (Eq.1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53bc710cf3779357c030fecd81c7bf815aae4396)
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![{\displaystyle \displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5daeea09b24684e26ca8bc6461a82f13ce9beb49)
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Find the integral of EQ 1 using integration by parts for n=0 and n=1
For n=0
For n=1
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![{\displaystyle \displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5daeea09b24684e26ca8bc6461a82f13ce9beb49)
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Take u=(x+1) and du=dx
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![{\displaystyle \displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5daeea09b24684e26ca8bc6461a82f13ce9beb49)
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This gives two separate integrals:
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![{\displaystyle \displaystyle (Eq.2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/549b9b71fb3dd781fd437c65c44e3b0bff3821db)
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Integrate the first integral (left hand) by parts, taking:
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![{\displaystyle \displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5daeea09b24684e26ca8bc6461a82f13ce9beb49)
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Giving:
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![{\displaystyle \displaystyle (Eq.3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67029a61096aa9304e425bf080fbeedf36390330)
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Integrate the second integral (right hand) by parts, taking: :
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![{\displaystyle \displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5daeea09b24684e26ca8bc6461a82f13ce9beb49)
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Giving:
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![{\displaystyle \displaystyle (Eq.4)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/698bb882efb26bc61f16ec2c4a36578d59e90561)
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Substituting EQ.3 and EQ.4 into EQ.2 and integrating the remaining integrals gives:
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![{\displaystyle \displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5daeea09b24684e26ca8bc6461a82f13ce9beb49)
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Since u=x+1
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![{\displaystyle \displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5daeea09b24684e26ca8bc6461a82f13ce9beb49)
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Giving the final answer of:
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![{\displaystyle \displaystyle (Eq.5)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/285fe6908ded39d6692cb8e1660f47e457751d81)
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