query: Justifying the integral identity “$\int \mathrm{e}^{cx} \sin bx\, \mathrm{d}x = \frac{\mathrm{e}^{cx}}{b^2 + c^2}(c \sin bx - b \cos bx)$”
positive_passages.docid: 22099
positive_passages.text: Integrate $e^{ax}\sin(bx)?$
negative_passages.docid: 152339
negative_passages.text: Complex-valued Fourier integral: $ \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $
negative_passages.docid: 31063
negative_passages.text: Evaluating $\int_0^2(\tan^{-1}(\pi x)-\tan^{-1} x)\,\mathrm{d}x$
negative_passages.docid: 133990
negative_passages.text: How prove this $\int_{a}^{b}f(x)dx=\frac{1}{2}(b-a)[f(a)+f(b)]-\frac{1}{12}(b-a)^3f''(\xi)$
negative_passages.docid: 48394
negative_passages.text: Proving bessel identity $[J_0(x)]^2 + 2[J_1(x)]^2 + 2[J_2(x)]^2 + \cdots = 1$ and $|J_0(x)|\le 1$
negative_passages.docid: 45725
negative_passages.text: Derivation of an identity with beta function.
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: 122481
negative_passages.text: proof of an identity for exponential function
negative_passages.docid: 105619
negative_passages.text: Finding $\int_{0}^{\pi/2} \frac{\tan x}{1+m^2\tan^2{x}} \mathrm{d}x$
negative_passages.docid: 28771
negative_passages.text: Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $
negative_passages.docid: 76408
negative_passages.text: Show that $\int_0^\pi f(\sin x)\,\mathrm{d}x = 2\int_0^{\pi/2}f(\sin x) \, \mathrm{d}x$
negative_passages.docid: 6852
negative_passages.text: Integral of $\int \frac{x \sin{x} \cos{x}}{(a^2 \sin^2 x + b^2 \cos^2 x)^2} dx $
negative_passages.docid: 4287
negative_passages.text: Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$
negative_passages.docid: 49306
negative_passages.text: Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x}\mathrm dx$
negative_passages.docid: 51991
negative_passages.text: What is the integral of $e^{\cos x}$
negative_passages.docid: 6573
negative_passages.text: Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$
negative_passages.docid: 13747
negative_passages.text: Proof of $\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}$