Integral using table and substiution (simple pics provided)

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January 22nd 2013, 09:27 PM
bikerboy2442

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Integral using table and substiution (simple pics provided)

Attachment 26671I'm supposed to solve this using an integral table and substitution.

I attempted to solve by rewriting the denominator as Sqrt((8x)²+9²) then using Attachment 26672 to solve, substituting u=8x.

Thereby getting the answer ln|8x+sqrt((8x)²+9²)|+C however this is apparently not correct. Any help?

*Edit: My provided integral table can be found here
January 22nd 2013, 10:18 PM
Prove It

Re: Integral using table and substiution (simple pics provided)

I would do the substitution $\displaystyle \begin{align*} x = \frac{9}{8}\sinh{t} \implies dx = \frac{9}{8}\cosh{t}\,dt \end{align*}$ and the integral becomes

$\displaystyle \begin{align*} \int{\frac{1}{\sqrt{64x^2 + 81}}\,dx} &= \int{\frac{1}{\sqrt{64\left( \frac{9}{8}\sinh{t} \right)^2 + 81}}\cdot \frac{9}{8}\cosh{t}\,dt} \\ &= \frac{9}{8}\int{\frac{\cosh{t}}{\sqrt{64 \left( \frac{81}{64}\sinh^2{t} \right) + 81 }}\,dt} \\ &= \frac{9}{8}\int{\frac{\cosh{t}}{\sqrt{81\sinh^2{t} + 81}}\,dt} \\ &= \frac{9}{8}\int{\frac{\cosh{t}}{\sqrt{81 \left( \sinh^2{t} + 1 \right) }}\,dt} \\ &= \frac{9}{8}\int{\frac{\cosh{t}}{\sqrt{81\cosh^2{t} }}\,dt} \\ &= \frac{9}{8}\int{\frac{\cosh{t}}{9\cosh{t}}\,dt} \\ &= \frac{1}{8}\int{1\,dt} \\ &= \frac{1}{8}t + C \\ &= \frac{1}{8}\sinh^{-1}{\left( \frac{8x}{9} \right)} + C \end{align*}$
January 23rd 2013, 03:19 AM
tom@ballooncalculus

Re: Integral using table and substiution (simple pics provided)

Note that Prove It's answer in log form (Inverse hyperbolic function - Wikipedia, the free encyclopedia) is exactly an eighth of yours after removing the constants.

All you missed was to divide your answer by 8 to adjust for the multiplying by eight caused by the chain rule 'on the way down', i.e. for differentiation.

The conventional way, of course, is to substitute 1/8 du for dx.

But, just in case a picture helps...

http://www.ballooncalculus.org/draw/.../thirtytwo.png

... where (key in spoiler) ...

Spoiler:

http://www.ballooncalculus.org/asy/chain.png

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

The general drift is...

http://www.ballooncalculus.org/asy/maps/intChain.png

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