Intereting Posts

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Accurate identities related to $\sum\limits_{n=0}^{\infty}\frac{(2n)!}{(n!)^3}x^n$ and $\sum\limits_{n=0}^{\infty}\frac{(2n)!}{(n!)^4}x^n$
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Does $G\times K\cong H\times K$ imply $G\cong H$?
For what values does this method converge on the Lambert W function?
Tips for finding the Galois Group of a given polynomial
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Proving completeness of a metric space
Given $A\cap \overline{B}\neq \emptyset$, prove $A\cup B$ connected.
If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$
Proof: $\mathrm{adj}(\mathrm{adj}(A)) = (\mathrm{det}(A))^{n-2} \cdot A$ for $A \in \mathbb{R}^{n\times n}$
Prove that a separable metric space is Lindelöf without proving it is second-countable

Evaluate the following integral

$$

\int \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a-x}} \, dx

$$

I just can’t seems to know what to do I have tried squaring out but it only gets worse.

- Fractional Trigonometric Integrands
- Solving this integral $\int\frac{1}{1+x^n} dx$?
- Derivating in respect of a function that's a derivative of another function
- What does a “half derivative” mean?
- Reduction Integration of $I_n=\int{\frac{x^n}{\sqrt{x^2+a^2}} dx}$
- Evaluate $\sum_{k=1}^\infty \frac{k^2}{(k-1)!}$.
- Where is the flaw in this “proof” that 1=2? (Derivative of repeated addition)
- Continued fraction for $\tan(nx)$
- Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$
- Euler-Maclaurin Summation Formula for Multiple Sums

Decompose the integrand:

$$

\frac{\sqrt{x}}{\sqrt{a-x}+\sqrt{x}} =

\frac{1}{2} +

\frac{a}{2 (2 x-a)}-

\frac{\sqrt{x(a-x)}}{2 x-a}

$$

You will find

$$

\int \frac{1}{(2 x-a)} \, dx =

\frac{1}{2} \ln \left( 2x – a \right)

$$

and

$$

\int \frac{\sqrt{x(a-x)}}{2 x-a} \, dx

= \frac{1}{2} \sqrt{x(a-x)}-\frac{1}{2} a \tanh ^{-1}\left(\frac{\sqrt{x}}{\sqrt{a-x}}\right)

$$

When successful, you will have

$$

\int \frac{\sqrt{x}}{\sqrt{a-x}+\sqrt{x}} \, dx =

\frac{1}{4} \left(2 x -2 \sqrt{x(a-x)}+a \ln (2 x-a)+2 a \tanh ^{-1}\left(\frac{\sqrt{x}}{\sqrt{a-x}}\right)\right)

$$

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- Determining k: $\int_{6}^{16} \frac{dx}{\sqrt{x^3 + 7x^2 + 8x – 16}} = \frac{\pi}{k}$
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- Commuting matrices and simultaneous diagonalizability