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Question Number 218781 Answers: 1 Comments: 0

prove: ∫_0 ^(π/4) arccos ((√2)/( (√(3−tan^2 x)))) dx=(π^2 /(24))

$$\mathrm{prove}: \\ $$$$\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\mathrm{arccos}\:\frac{\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{3}−\mathrm{tan}^{\mathrm{2}} \:{x}}}\:{dx}=\frac{\pi^{\mathrm{2}} }{\mathrm{24}} \\ $$

Question Number 218780 Answers: 2 Comments: 1

Question Number 218779 Answers: 3 Comments: 0

Question Number 218778 Answers: 4 Comments: 0

Question Number 218771 Answers: 1 Comments: 0

∫_0 ^( ∞) J_ν (αt)J_ν (βt)dt=?? (α,β≠0)

$$\int_{\mathrm{0}} ^{\:\infty} \:{J}_{\nu} \left(\alpha{t}\right){J}_{\nu} \left(\beta{t}\right)\mathrm{d}{t}=?? \\ $$$$\left(\alpha,\beta\neq\mathrm{0}\right) \\ $$

Question Number 218775 Answers: 2 Comments: 0

Question Number 218767 Answers: 1 Comments: 0

Σ_(ℓ∈(−∞,∞)) J_ℓ (z)=??

$$\underset{\ell\in\left(−\infty,\infty\right)} {\sum}\:{J}_{\ell} \left({z}\right)=?? \\ $$

Question Number 218766 Answers: 0 Comments: 0

Σ_(ℓ=1) ^∞ (1/ℓ)J_ν (ℓt)=??

$$\underset{\ell=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\ell}{J}_{\nu} \left(\ell{t}\right)=?? \\ $$

Question Number 218765 Answers: 1 Comments: 0

solve ∫_0 ^( ∞) J_ν (kt)e^(−wt) dt=g_(ν,k) (w)

$$\mathrm{solve} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\:{J}_{\nu} \left({kt}\right){e}^{−{wt}} \mathrm{d}{t}=\mathrm{g}_{\nu,{k}} \left({w}\right) \\ $$

Question Number 218749 Answers: 1 Comments: 0

Question Number 218748 Answers: 1 Comments: 0

∫_0 ^∞ ((x^2 cos x)/(cosh 2x−cos x))−((2x^2 )/(e^(4x) −2e^(2x) cos x+1))dx,lemma:Σ_(k=1) ^(+∞) ((cos kx)/p^k )=((p cos x−1)/(p^2 −2p cos x+1)),p>1

$$\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{2}} \mathrm{cos}\:{x}}{\mathrm{cosh}\:\mathrm{2}{x}−\mathrm{cos}\:{x}}−\frac{\mathrm{2}{x}^{\mathrm{2}} }{\mathrm{e}^{\mathrm{4}{x}} −\mathrm{2}{e}^{\mathrm{2}{x}} \mathrm{cos}\:{x}+\mathrm{1}}{dx},\mathrm{lemma}:\underset{{k}=\mathrm{1}} {\overset{+\infty} {\sum}}\frac{\mathrm{cos}\:{kx}}{{p}^{{k}} }=\frac{\mathrm{p}\:\mathrm{cos}\:{x}−\mathrm{1}}{{p}^{\mathrm{2}} −\mathrm{2}{p}\:\mathrm{cos}\:{x}+\mathrm{1}},{p}>\mathrm{1} \\ $$

Question Number 218739 Answers: 2 Comments: 0

Question Number 218738 Answers: 2 Comments: 0

Question Number 218737 Answers: 2 Comments: 0

Question Number 218736 Answers: 5 Comments: 0

Question Number 218735 Answers: 5 Comments: 0

Question Number 218734 Answers: 7 Comments: 0

Question Number 218733 Answers: 4 Comments: 1

Question Number 218719 Answers: 0 Comments: 0

Question Number 218713 Answers: 1 Comments: 3

Question Number 218709 Answers: 2 Comments: 0

∫_0 ^(π/2) (dθ/( (√(sinθ +cosθ))))

$$ \\ $$$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \:\frac{{d}\theta}{\:\sqrt{{sin}\theta\:+{cos}\theta}} \\ $$$$ \\ $$

Question Number 218703 Answers: 3 Comments: 0

Prove; lim_(x→0) ((x − sin x)/x^3 ) = (1/6)

$$ \\ $$$$\:\:\:\:{Prove};\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{x}\:−\:{sin}\:{x}}{{x}^{\mathrm{3}} }\:=\:\frac{\mathrm{1}}{\mathrm{6}}\:\: \\ $$$$ \\ $$

Question Number 218676 Answers: 2 Comments: 0

Question Number 218675 Answers: 4 Comments: 0

Question Number 218674 Answers: 3 Comments: 0

Question Number 218673 Answers: 3 Comments: 1

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