# Questions tagged [substitution]

Questions that involve a replacement of variable(s) in an expression or a formula.

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### Making the substitution $x=e^y$ in the integral $\int_0^1 x^{2n} \ln x/(1+x^2) dx$

$$\mbox{The integral}\quad \int_{0}^{1}x^{2n}\frac{\ln\left(x\right)}{1 + x^{2}}{\rm d}x\quad \mbox{converges to}\quad r_{n} + {\rm C}s_{n}$$ when $n$ is a non negative integer. Where $r_{n},s_{n}$ ...
• 2,691
1 vote
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### How do we solve the ODE $(2y+x)y' + 2 = y^2 + xy - y$ according to the substitution $v = y^{2} + yx$?

I am attempting to perform the following substitution in the non-linear ODE below: $$v=y^2+yx$$ $$(2y+x)y'+2=y^2+xy-y$$ After attempting a multitude of different approaches, I still am unable to ...
• 23
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### Integration measure for a strange substitution

I have a 2D integral over a momentum vector, i.e. $\int dp_x dp_y$ and the substitution for this is given by $$\xi = |\vec{p}| + |\vec{p} + \vec{q}| , \, \, \, \eta = |\vec{p}| - |\vec{p} + \vec{q}|$$...
1 vote
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### How to calculate $\int_{0}^{2\pi}(a+\sin{x})^{-3/2}dx$ [closed]

I wonder if there is an analytic solution for the following equation: $$\int_{0}^{2\pi}(a+\sin{x})^{-3/2}dx$$ Here, $a$ is a constant. Would you please give an advice?
• 37
1 vote
152 views

### $\int_0^R \frac{r^{l+1}}{\sqrt{R^2 - r^2}}\text dr$

I'm trying to solve problem 3.3 from Jackson's Classical Electrodynamics, but I'm encountering some troubles solving $$\int_0^R \frac{r^{l+1}}{\sqrt{R^2 - r^2}}\text dr, \qquad l = 0,2,4,6,\ldots$$ ...
• 135
103 views

### Solve $\Bigl(2 \sin(x)D^2+2 (\cos(x)+\sin(x))D\Bigr)y+2y\cos(x)=\cos(x)$

How to solve the following differential equation? $$2 \sin(x)\frac{d^2 y}{dx^2}+2 \cos(x) \frac{dy}{dx}+2 \sin(x) \frac{dy}{dx}+2y \cos(x)=\cos(x)$$ This was given to me by one of my friends as a ...
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### Problem with simplification of this indefinite integral

Let's consider this fairly easy problem. $$\int \sin x \cos x dx$$ which can be transformed into $$\dfrac12 \int \sin 2x \ dx$$ Setting $t = 2x$, we have $dt/dx = 2$ Substituting this value into our ...
• 1
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### differential equations, substitution suggested by the equation

I took my examination in differential equations earlier; i would've gotten a perfect score, but i tripped in this problem: $$(1+5y\sin x)dy + y^4\cos xdx = 0$$ I used substitution suggested by the ...
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• 25.2k
1 vote
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### Definite integral substitution problem

I had to solve the following definite integral, i solved it by substitution but I checked many times and I can't find out why the result I get is wrong. I know it can be solved by parts, but I want to ...
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I'm trying to teach myself simple electronic formulas. I haven't done algebra in awhile, and I'm very rusty. I know that $P = iV$, and $V = iR$, therefore, $P = i^2R$. How could I figure out that $P = ... • 1,661 0 votes 0 answers 69 views ### Generalizing an identity for polynomials [duplicate] For any quadratic function, prove that: $$f(x+3)-f(x) = 3[f(x+2)-f(x+1)]$$ This was a fairly straightforward problem, I solved it by assuming$f(x)=ax^2 + bx + c$and by simplifying$g(x)=f(x+3)-3f(...
• 739
Suppose we have a two dimensional rational function $R(x,y)$. Then we can use Weierstrass substitution for following integral: $$\int R(\sin(x), \cos(x)) \, dx .$$ But can we use Weierstrass ...