\[\large \int_{0}^{\infty} \frac{e^{-x} \sin(x)\cos(x)}{\sqrt{x}}\,dx = \frac{\sqrt{\pi}}{2\cdot \sqrt[4]{5}} \cdot \sin\left(\frac{1}{2} \tan^{-1}(2)\right)\]
It is trivial to prove the equation above using DogTeX, but can you prove it without DogTeX?
This is a part of the set Formidable Series and Integrals
Easy Math Editor
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Let I=∫0∞xe−xsin(x)cos(x)dx
=21∫0∞xe−xsin(2x)dx
=4i1∫0∞xe−x(e2ix−e−2ix)dx
=4iA−A∗
=21ℑ(A)
where A=∫0∞xe−x(1−2i)dx
Note that,
Γ(t)=∫0∞xt−1e−xdx
Substituting x↦ax, we have,
Γ(t)=at∫0∞xt−1e−axdx
⟹A=Γ(21)1−2i1
⟹I=2πℑ(1−2i1)
=245π⋅sin(21tan−1(2))
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Wonderful work as usual! +1
@Ishan Singh good ishu... :)
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So cute!!!!
Marvellous solution! +1
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LOL!