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I recently stumbled upon the following reduction formula on the internet which I am so far unable to prove.
$$I_n=\int\frac{\mathrm{d}x}{(ax^2+b)^n}\\I_n=\frac{x}{2b(n-1)(ax^2+b)^{n-1}}+\frac{2n-3}{2b(n-1)}I_{n-1}$$
I tried the substitution $x=\sqrt{\frac ba}t$, and it gave me
$$I_n=\frac{b^{1/2...
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Q:Integration of $e^{ax}\cos^n bx$ and $e^{ax}\sin^n bx$
I know how to integration $e^{ax}\cos bx$ using $\cos bx=\frac{e^{ibx}+e^{-ibx}}{2}$.Using the same trick here I got $$\int e^{ax}\cos^n bx dx=\int e^{ax}\left(\frac{e^{ibx}+e^{-ibx}}{2}\right)^n$$But it doesn't look easy to solve.I go...
Integration by reduction formula in integral calculus is a technique or procedure of integration, in the form of a recurrence relation. It is used when an expression containing an integer parameter, usually in the form of powers of elementary functions, or products of transcendental functions and polynomials of arbitrary degree, can't be integrated directly. But using other methods of integration a reduction formula can be set up to obtain the integral of the same or similar expression with a lower integer parameter, progressively simplifying the integral until it can be evaluated. This method...
I mean why was a separate term "continued proportion" created simply if $$a:b::b:d$$ Why could it just not have been called proportion?
Is there any bigger use of continued proportion at a bigger level hence a separate term "continued proportion" was given to it?
