query: Why is the antiderivative of $\frac{1}{1+x^2}=\tan^{-1}(x)$?
positive_passages.docid: 146594
positive_passages.text: Prove that $\frac{d}{dx}(\tan^{-1}(x))=\frac{1}{1+x^2}dx$
negative_passages.docid: 138914
negative_passages.text: Reduction formula for $\int \tan^n x dx$
negative_passages.docid: 7296
negative_passages.text: Prove $(1-\cos x)/\sin x = \tan x/2$
negative_passages.docid: 128355
negative_passages.text: Does $e^{tx} + x = t - 1$ yield a solution/s to $\frac{dx}{dt} = \frac{e^{-tx} - x}{e^{-tx} + t}$?
negative_passages.docid: 84597
negative_passages.text: Derivative of $(1-x)^{-2}$
negative_passages.docid: 70765
negative_passages.text: If $\frac{\tan(a+b-c)}{\tan(a-b+c)}=\frac{\tan c}{\tan b}$, then $\sin(b-c)=0$ or $\sin 2a + \sin2b + \sin 2c=0$
negative_passages.docid: 17769
negative_passages.text: Show that $\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$
negative_passages.docid: 154172
negative_passages.text: Finding $\tan t$ if $t=\sum\tan^{-1}(1/2t^2)$
negative_passages.docid: 116504
negative_passages.text: Interpolating an Antiderivative
negative_passages.docid: 21088
negative_passages.text: Prove $\int\limits_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$
negative_passages.docid: 108923
negative_passages.text: What is wrong with this false proof? $-1=(-1)^1=(-1)^\frac{2}{2}=((-1)^2)^\frac{1}{2}=\sqrt{(-1)^2}=\sqrt{1}=1$
negative_passages.docid: 33905
negative_passages.text: Prove the following trigonometric identity: $\frac{\tan A+\sec A-1}{\tan A-\sec A+1}=\frac{1+\sin A}{\cos A}$
negative_passages.docid: 48956
negative_passages.text: Simplify $\arctan (\frac{1}{2}\tan (2A)) + \arctan (\cot (A)) + \arctan (\cot ^{3}(A)) $
negative_passages.docid: 153867
negative_passages.text: Evaluate $ \int_{0}^{\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\:\mathrm dx$
negative_passages.docid: 36128
negative_passages.text: Using differentiation for computing $\int_{0}^{1}\frac{\arctan(ax)}{x\sqrt{1-x^2}}\,dx$
negative_passages.docid: 58909
negative_passages.text: Evaluation of $\int\frac{dx}{x+ \sqrt{x^2-x+1}}$
negative_passages.docid: 63191
negative_passages.text: What is the $k-$th derivative of $ \frac{1}{1-x^2}$?