Mathxx
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Complex number $\tan \alpha+i$
3 votes

I figured it out. $1^2+(\tan \alpha)^2=(\sec \alpha)^2$ So $\left |z \right |=\sec \alpha$ Is it right?

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Area of an ellipse. (Calculus)
3 votes

$$\frac{x^2}{4}+\frac{(y-3)^2}{9}=1$$ Let $Y=y-3$ So, \begin{align*} \frac{x^2}{4}+\frac{Y^2}{9} & = 1\\ \frac{x^2}{4} & =1-\frac{Y^2}{9}\\ \frac{x^2}{4} & =\frac{1}{9}(9-Y^2)\\ x^2 &...

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Solve$\int\frac{x^4}{1-x^4}dx$
2 votes

By doing long division, you'll get $$-\int \left(-\frac{1}{2(x^2+1)}-\frac{1}{4(x+1)}+\frac{1}{4(x-1)}+1\right) dx$$

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Find the values of m and n(Trigononetry in series)
2 votes

Found a way to solve from another site. $$S=\frac{1}{8}\left(3\sum_{k=1}^{44}\left(\cos\left(4k\frac{\pi}{180}\right)\right)+221\right)$$ $$\sum_{k=1}^{44} \left( \cos \left(4k \frac{ \pi}{180} \...

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Combined data of probability
1 votes

As @AlexR said, $P(\text{dog} \cap \neg \text{cat}) = P(\text{dog}) - P(\text{dog} \cap \text{cat})$ So you'll get 0.1 for P(Dog and not cat). $P(\text{Not cat} \mid \text{Dog})=\frac{0.1}{0.25}=0.4$...

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Mean and Variance (Without calculator)
1 votes

$\bar{y}=\frac{\sum y}{n}=\frac{0}{11}=0$ $\sigma y=\sqrt{\frac{1.2}{11}-0^2}$ $y=x-1506.8$ So, $\bar{y}=\bar{x}-1506.8$ $0=\bar{x}-1506.8$ $\bar{x}=1506.8$ $\sigma y=\sigma x$ $\sigma x=\sqrt{\...

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Are these two answers to $\int \sin ^3(x)dx$ equivalent?
1 votes

I found an alternative way. $$\cos 3x=\cos (2x+x)=\cos (2x)\cos (x)-\sin (2x)\sin (x)$$ $$=(2\cos ^2x-1)\cos x-(2\sin x\cos x)\sin x$$ $$=2\cos ^3x-\cos x-2\sin ^2x\cos x$$ $$=2\cos ^3x-\cos x-2(1-\...

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Arithmetic sequence.
0 votes

The proposed solution by the author. The common difference of the given sequence is $7$, and there are $21$ terms in the sequence, since $\frac{177-37}{7}=20$. Let $n$ be a sum of five terms in the ...

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What is a good book to learn about pre algebra?
0 votes

If you're saying the most basic algebra. Then I would suggest this. Introduction to Algebra

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reference-Request-IMO
0 votes

https://artofproblemsolving.com/wiki/index.php?title=Math_books Try this links. Books which specific in every level according to the level.

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Derivative of $\sqrt{ \left | x\right | } $
0 votes

By using the chain rule. $\frac{d}{dx}\sqrt{\left | x \right |}$ $=\frac{\frac{d}{dx}(\left |x \right |)}{2\sqrt{\left | x \right |}}$ $=\frac{x}{\left |x \right |}\frac{1}{2\sqrt{\left | x \right ...

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Perimeter of Triangle ABC
0 votes

I've found the answer by using Law of Cosines. $BC^2=8^2+8^2-2(8)(8)\cos 120=192$ $BC=8\sqrt{3}$ $AB+BC+AC=24+8\sqrt{3}$

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The partial fraction decomposition of $\dfrac{x}{x^3-1}$
0 votes

$\frac{x}{x^3-1}=\frac{x}{(x-1)(x^2+x+1)}$ $\frac{x}{(x-1)(x^2+x+1)}=\frac{A}{x-1}+\frac{Bx+C}{x^2+x+1}$ $x=A(x^2+x+1)+(Bx+C)(x-1)$ Let $x=1$ $1=A(1^2+1+1)$ $1=3A$ $A=\frac{1}{3}$ So, $x=\frac{...

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First Principles of differentiation.
0 votes

$$f'(x)=\lim_{{h}\to{0}}\frac{(x+h)^{x+h}-x^x}{h}$$ $$=\lim_{{h}\to{0}}\frac{(x+h)^{x+h}-(x+h)^x+(x+h)^x-x^x}{h}$$ $$=\lim_{{h}\to{0}}\frac{(x+h)^x\left[(x+h)^h-1\right]}{h}+\lim_{{h}\to{0}}\frac{(x+...

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Integration of logarithm
0 votes

$$I = \int \ln(\ln \sqrt{x})^{\ln x} dx$$ $$ = \int \ln \left(\ln \left(x^{1/2}\right) \right)^{\ln x} dx$$ $$ = \int \ln \left(\frac{1}{2}\ln x \right)^{\ln x} dx$$ $$= \int (\ln x) \cdot \ln \left(\...

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