# Questions tagged [substitution]

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

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### Is it possible to turn this 'proof' of the product rule into a rigorous argument?

I have often found linear approximation to be useful in understanding the main theorems of calculus. I tried using it to 'prove' the product rule, as I find the typical proof for it to be unintuitive. ...
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### L'Hospital rule to find limit on indeterminate form.

I have the following limit to compute: $$\lim_{x\to 0}{\left(\cos x -1\over {5 x^2}\right)}$$ I need to find the limit as $x \to 0$. I tried using L'Hospital rule , so I found the derivative of the ...
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### Solve $\int \frac{5}{4x^2+3}dx$

I've tried few way to resolve $\int \frac{5}{4x^2+3}dx$ but I think there's somthing I'm missing. As a first step I've took the constant out: $5\int \frac{1}{4x^2+3}dx$. Next I've thought it would be ...
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### How to integrate with given substitution?

Using the substitution $u = x^2e^{-4x} + 3$, find $$\int\frac{x(1-2x)}{x^2+3e^{4x}} dx$$
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### How does a polynomial transform upon making a substitution into it?

My title is very poorly worded, I apologise, I'm having a hard time wording the question. My lecturer described it as being obvious, so I must be missing something very fundamental (the substitution ...
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### Integrate $\ln(\sqrt{1-x}+\sqrt{1+x})$

Integrate $$\ln(\sqrt{1-x}+\sqrt{1+x})$$ My attempts: i) Multiplying and dividing its conjugate, we get $$I=\int \ln(-2x)dx-\int\ln(\sqrt{1-x}-\sqrt{1+x})dx$$ The second term is equally challenging ...
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### Integral with substitution and probability distribution function

I have the following integral: $$\int_{0}^{n} (a x+b) g(x) dx$$ $g(x)$ is a probability density function. and $x= \epsilon + c$ where $c$ is deterministic and $\epsilon \sim f$. I want to rewrite ...
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### Help with substitution

I am having troubles with the following transformation. I have: \begin{align*} Y_t^k=\int_0^t(t-s)^{\alpha-1}h(s)ds+\int_0^t (t-s)^{\alpha-1}\int_0^s(s-u)^{-\alpha}\sigma_{n_k}(X_u^{n_k})dB_u^{n_k}ds \...
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### Different answers when changing variables in a definite integral with an infinite lower limit

I've searched for a post about this, to no avail. I am pretty sure I know which method is correct, but I'm having trouble pinning down exactly why the incorrect method is wrong. I wish to evaluate the ...
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### Substitution inside integral?

Let $X$ be a random variable. I am trying to prove a special case of LOTUS (assuming $g$ is increasing and differentiable) using transformation of PDF. I have in my proof arrived at the following ...
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