query: $\int \sin x \cos x \ dx \neq \frac {\sin^2x}{2}$ if we use the identity $\sin A \cos B = \frac12[\sin(A-B) + \sin(A+B)]?$
positive_passages.docid: 31922
positive_passages.text: Integral of $\sin x \cos x$ using two methods differs by a constant?
negative_passages.docid: 130736
negative_passages.text: Integrating $\int^0_\pi \frac{x \sin x}{1+\cos^2 x}$
negative_passages.docid: 48586
negative_passages.text: Prove: $\cot x=\sin x\,\sin\left(\frac{\pi}{2}-x\right)+\cos^2x\,\cot x$
negative_passages.docid: 65093
negative_passages.text: Integrate $\int \sin^4x \cos^2x dx$
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: 40556
negative_passages.text: Why is $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$
negative_passages.docid: 70715
negative_passages.text: How do I prove $\int_0^\pi\frac{(\sin nx)^2}{(\sin x)^2}dx = n\pi$?
negative_passages.docid: 54175
negative_passages.text: Prove the trigonometric identity $\sin^4{x} = \frac{3-4\cos{2x}+\cos{4x}}{8}$
negative_passages.docid: 123899
negative_passages.text: $\int_0^{+\infty}\frac{\sin(x)}{x^\alpha}dx=\frac{\Gamma(\alpha/2)\Gamma(1-\alpha/2)}{2\Gamma(\alpha)}$
negative_passages.docid: 84683
negative_passages.text: integrate $dx/(a^2 \cos^2x+b^2 \sin^2x)^2$
negative_passages.docid: 137376
negative_passages.text: Proving a trig identity: $\frac{\sin(A + B)}{\sin(A - B)}=\frac{\tan A + \tan B}{\tan A - \tan B}$
negative_passages.docid: 20016
negative_passages.text: How to integrate $\int_0^{\pi/2}\frac{\sin^nx}{\sin^nx+\cos^nx}dx$?
negative_passages.docid: 123500
negative_passages.text: Integration of $\frac {1}{1+\sin x}$
negative_passages.docid: 6573
negative_passages.text: Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$
negative_passages.docid: 7296
negative_passages.text: Prove $(1-\cos x)/\sin x = \tan x/2$
negative_passages.docid: 89387
negative_passages.text: Integrate : $\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}dx$
negative_passages.docid: 29370
negative_passages.text: Trig integral $\int ( \cos{x} + \sin{x}\cos{x}) \, dx $