Articles of substitution

Can indefinite double integrals be solved by change of variables technique?

I have a simple double integral: $$\int\int(x+y)dxdy$$ Now, change the variables according to the following scheme: $$x=u+v$$ $$y=u-v$$ The jacobian is then: $$|\frac{\partial (x,y)}{\partial (u,v)}|=1$$ Substitute u and v into the integral, solving the integral and substituting y and x back in their place equals: $$(\frac{x-y}{2})(\frac{x+y}{2})^2$$ Which is obviously not the same as the result […]

Very confused about a limit.

This question is about where I made my mistake in the computation of a limit. It relates to An answer I gave that confused me. The question to which I gave the (partial) answer is related to tetration but my mistake is probably a simple general one ( considering tetration as complicated ). Here is […]

On how to solve the ODE $v'-\frac{1}{v}=x$…

I’ve been having trouble finding the general solution of $v$ for $v’-\frac{1}{v}=x$. I’ve attempted various substitutions in attempts of obtaining separation of variables or recognizable form to apply the method of the integrating factor. A couple of substitutions I’ve attempted: $$\alpha=\frac{1}{v}$$ $$\beta=\frac{1}{\alpha^2}$$ I tried others but threw out the scratch paper (yeah…would’ve helped now to […]

Integration by substitution gone wrong

I’ve noticed that using integration by substitution blindly could lead to some strange results. For example: with $u = x^2$, we might naively follow the usual procedure to find $$ \int_{-1}^1 x^4\,dx = \int_{-1}^{1} x^3 \,x\,dx = \int_{u(-1)}^{u(1)} u^{3/2}\,du = \int_1^1 u^{3/2}\,du =0 $$ The incorrect step here is writing $x = u^{1/2}$, since we […]

Indefinite Integral of Floor Function Integration by Substitution

From knowing the anti-derivative of floor function to be x*floor(x), is it possible to find the derivative of a function contained within a floor function? The particular question I had in mind is floor(y(x)/17) and I believe in the same way y in an equation can be treated as y function of x, I tried […]

A inequality proposed at Zhautykov Olympiad 2008

A inequality proposed at Zhautykov Olympiad 2008 Let be $a,b,c >0$ with $abc=1$. Prove that: $$\sum_{cyc}{\frac{1}{(a+b)b}} \geq \frac{3}{2}.$$ $a=\frac{x}{y}$, $b=\frac{y}{z}$, $c=\frac{z}{x}$. Our inequality becomes: $$\sum_{cyc}{\frac{z^2}{zx+y^2}} \geq \frac{3}{2}.$$ Now we use that: $z^2+x^2 \geq 2zx.$ $$\sum_{cyc}{\frac{z^2}{zx+y^2}} \geq \sum_{cyc}{\frac{2z^2}{z^2+x^2+2y^2}} \geq \frac{3}{2}.$$ Now applying Cauchy Schwarz we obtain the desired result . What I wrote can be found […]

Recurrence relation using substitution method

How do I solve the following recurrence using substitution method? $$T(n) = T(n-1)+C$$ I’ve found reference to so many examples on line but most of the examples are of the form $$T(n) = T\left(\frac{n}{4}\right) + n$$ So, I am really struggling to understand it and to solve $T(n) = T(n-1)+C$ using substitution method. I appreciate […]

Trigonometric substitution and triangles

I’m learning trigonometric substitutions and am having a bit of trouble understanding the intuition behind the conversions (why do most use secant?). If you could explain the conversions geometrically using a triangle, that would be very helpful. For example, if we have $$\int \frac{\sqrt{x^2-4}}{x}\,dx$$ I tried to construct a triangle like so: To get $$\sin(\theta)=\frac{\sqrt{x^2-4}}{x}$$ […]

determining an integral using only derivative properties of two functions

let $\alpha'(x)=\beta(x), \beta'(x)=\alpha(x)$ and assume that $\alpha^2 – \beta^2 = 1$. how would I go about calculating the following anti derivative : $\int (\alpha (x))^5 (\beta(x))^4$d$x$. I have tried many different methods with no success. Firstly, I tried to write the integral in terms of one of the variables (using the first condition) and then […]

How can I solve $\int \frac{1}{\sqrt{x^2 + 7x} + 3}\, dx$

How can I solve the following integral? $$\int \frac{1}{\sqrt{x^2 + 7x} + 3}\, dx$$ $$\begin{align} \int \frac{1}{\sqrt{x^2 + 7x} + 3}\, dx&=\int \frac{1}{\sqrt{\left(x+\frac72\right)^2-\frac{49}{4}} + 3}\, dx \end{align}$$ $$u = x+\frac{7}{2}, \quad a = \frac{7}{2}$$ $$\int \frac{1}{\sqrt{u^2 – a^2} +3}\, \,du$$ Attempt I – By Trigonometric substitution $$\sqrt{u^2 – a^2} = \sqrt{a\sec^2\varTheta – a^2} = \sqrt{a^2(\sec^2\varTheta […]