Hi everyone, I've been working on integration techniques for SM over the last week... and I've been stuck on some of the questions (this is not an SM book, so the questions are generally pitched harder than ordinary VCE).
}{(1+x^2)^2} dx)
I've tried substituting t = (1+x^2), but it doesn't seem to make any headway.
Another one is
}{\sqrt{1+x^2}} dx)
It looks like the above two are related or somehow consequential...
Any help would be greatly appreciated ^_^
Let's actually try and solve the op's question
(a)
There's a log...and I know I can integrate the bit of the expression without the log, so I'll integrate by parts.
Letting u = ln x and dv/dx = x/(1+x^2)^2, we have v = -1/(2(1+x^2)) and du/dx = 1/x
Hence the integral becomes uv - int v du = -ln x/(2(1+x^2)) + int(1/(2x(1+x^2)) dx)
This integral is a fairly standard integral to do by partial fractions
(b)
My gut instinct is to go with a substitution x = sinh t because the top contains an inverse hyperbolic sine (ln (x+sqrt(x^2+1)) = arsinh x) and the bottom is the derivative of arsinh x
So if x = sinh t, dx = cosh t dt = sqrt(x^2 + 1) dt
dt = dx/(sqrt(x^2+1))
Plugging everything in, we get integral of x ln(x+sqrt(x^2+1))/sqrt(x^2+1) dx = integral of sinh t*ln(sinh t + sqrt(sinh^2 t + 1) dt = int(sinh t * ln(sinh t + cosh t dt) = int(sinh t * ln(e^t) dt) = int(t sinh t) dt
This is another standard integral to do by parts.
My working won't make sense if you don't understand how to use hyperbolic functions. If you want, do the working with x = (e^t - e^-t)/2 instead
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