# Tagged Questions

63 views

### Definite integration of a trigonometric function

How to integrate $$\int_0^{\pi/2}\!\dfrac{2a \sin^2 x}{a^2 \sin^2 x +b^2 \cos^2 x}\,dx$$ my first step is $$\frac{2}{a} \int_0^{\pi/2}\!\dfrac{a^2 \sin^2 x}{a^2 +(b^2 - a^2) \cos^2 x}\, dx$$ I ...
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### $\int_0^\infty\text{Ci}(x)^3\mathrm dx$

Is there a closed form for this integral: $$\int_0^\infty\text{Ci}(x)^3\mathrm dx,$$ where $\text{Ci}(x)=-\int_x^\infty\frac{\cos z}{z}\mathrm dz$ is the cosine integral?
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### Evaluating a double integral involving exponential of trigonometric functions

I am having trouble evaluating the following double integral: $$\int\limits_0^\pi\int\limits_0^{2\pi}\exp\left[a\sin\theta\cos\psi+b\sin\theta\sin\psi+c\cos\theta\right]\sin\theta d\theta\, d\psi$$ ...
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### Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$

Prove that $$\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$$
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### Definite double integral of a trigonometric function

I am in trouble in the calculation of the following double integral: $$\int_0^a d\rho\int_0^{2\pi}d\phi\exp(-ik\rho(\sin(\theta_0)\cos(\phi_0-\phi)+\sin(\theta_1)\cos(\phi_1-\phi)))\rho$$ Many thanks
### Finding integral $\int_{- \pi}^{\pi} \cos(mt) \cos(\lambda t) dt$
I am a little stuck on the following problem: Prove that: $$\int_{- \pi}^{\pi} \cos(mt) \cos(\lambda t) dt = -2 \frac{(-1)^{m} \lambda \sin(\pi \lambda)}{m^2 - \lambda ^2}$$ I have used the fact ...