Intereting Posts

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proving Jensen's formula
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Find the positive $n$ such $2^{n-2i}|\sum_{k=i}^{\lfloor\frac{n-1}{2}\rfloor}\binom{k}{i}\binom{n}{2k+1}$

Express recurrence relation of the integral

$$

I_n=\int\frac{dx}{(1+x^2)^n}

$$

[My Answer]

- How to write an integral as a limit?
- Evaluating series of zeta values like $\sum_{k=1}^{\infty} \frac{\zeta(2k)}{k16^{k}}=\ln(\pi)-\frac{3}{2}\ln(2) $
- Computing $ \int_{0}^{\infty} \frac{1}{(x+1)(x+2)…(x+n)} \mathrm dx $
- Does $\int_{-1}^1\frac{\arctan x}{\text{arctanh}\,x}\,\mathrm{d}x$ have a closed form?
- Integration operation, and its relation to differentials
- Riemann sum of $\sin(x)$

$$

I_n = \int\frac{1+x^2}{(1+x^2)^n}dx-\int\frac{x^2}{(1+x^2)^n}dx

$$

$$

I_n=I_{n-1}-\int x\cdot\frac{x}{(1+x^2)^n}dx

$$

$$

I_n=I_{n-1}-\frac{x}{2(1-n)(x^2+1)^{n-1}}+\frac{1}{2(1-n)}I_{n-1}

$$

$$

I_n=\frac{2n-3}{2(n-1)}I_{n-1}+\frac{x}{2(n-1)(x^2+1)^{n-1}} \ \ \ \ (n>1)

$$

$$

I_1=\arctan(x)

$$

Is my answer correct?

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- Help finding a closed form

Yes, the answer is correct (up to a constant, but it does not not change the idea).

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