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show that ∫e^(sin(x)) dx= −Σ_(n=0) ^∞ (1/(n!))[ cos(x)∗(sin(x))^(n+1) ∗[(sin(x))^2 ]^((((−n)/2)−(1/2))) ∗ 2F_1 [(1/2),((1−n)/2);(3/2);(cos(x))^2 ] ]+c notice\2F_1 is special function called hypergeometric function

$${show}\:{that} \\ $$$$\int{e}^{{sin}\left({x}\right)} \:{dx}= \\ $$$$−\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}!}\left[\:{cos}\left({x}\right)\ast\left({sin}\left({x}\right)\right)^{{n}+\mathrm{1}} \ast\left[\left({sin}\left({x}\right)\right)^{\mathrm{2}} \right]^{\left(\frac{−{n}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}\right)} \ast\:\mathrm{2}{F}_{\mathrm{1}} \left[\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}−{n}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};\left({cos}\left({x}\right)\right)^{\mathrm{2}} \right]\:\right]+{c} \\ $$$$ \\ $$$${notice}\backslash\mathrm{2}{F}_{\mathrm{1}} \:{is}\:{special}\:{function}\:{called}\:{hypergeometric}\:{function} \\ $$

Question Number 87769 Answers: 2 Comments: 0

∫ ((ln(e^x +1))/(e^(−x) +1)) dx

$$\int\:\frac{\mathrm{ln}\left(\mathrm{e}^{\mathrm{x}} +\mathrm{1}\right)}{\mathrm{e}^{−\mathrm{x}} +\mathrm{1}}\:\mathrm{dx}\: \\ $$

Question Number 87757 Answers: 0 Comments: 2

∫_a ^b ((√(x−a))/(√(b−x))) dx =?

$$\int_{\mathrm{a}} ^{\mathrm{b}} \:\frac{\sqrt{\mathrm{x}−\mathrm{a}}}{\sqrt{\mathrm{b}−\mathrm{x}}}\:\mathrm{dx}\:=?\: \\ $$

Question Number 87723 Answers: 1 Comments: 0

∫((1/(x−1))+((Σ_(k=0) ^(2018) (k+1)x^k )/(Σ_(k=0) ^(2019) x^k )))dx

$$\int\left(\frac{\mathrm{1}}{{x}−\mathrm{1}}+\frac{\underset{{k}=\mathrm{0}} {\overset{\mathrm{2018}} {\sum}}\left({k}+\mathrm{1}\right){x}^{{k}} }{\underset{{k}=\mathrm{0}} {\overset{\mathrm{2019}} {\sum}}{x}^{{k}} }\right){dx} \\ $$

Question Number 87716 Answers: 1 Comments: 1

Question Number 87711 Answers: 1 Comments: 2

∫_0 ^∞ ((1−xe^(−x) −e^(−x) )/(x(e^x −e^(−x) )))dx

$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}−{xe}^{−{x}} −{e}^{−{x}} }{{x}\left({e}^{{x}} −{e}^{−{x}} \right)}{dx} \\ $$

Question Number 87709 Answers: 0 Comments: 0

sbow that ∫_1 ^∞ (([3x])/(([x])!))dx=4e−1

$${sbow}\:{that} \\ $$$$\int_{\mathrm{1}} ^{\infty} \frac{\left[\mathrm{3}{x}\right]}{\left(\left[{x}\right]\right)!}{dx}=\mathrm{4}{e}−\mathrm{1} \\ $$

Question Number 87692 Answers: 0 Comments: 8

sir Ma?h+t?que you have posted ∫(dx/(((x+1)....(x+n))^2 ))=......can you reposted it please

$${sir}\:{Ma}?{h}+{t}?{que}\:{you}\:{have}\:{posted} \\ $$$$\int\frac{{dx}}{\left(\left({x}+\mathrm{1}\right)....\left({x}+{n}\right)\right)^{\mathrm{2}} }=......{can}\:{you}\:{reposted}\:{it}\:{please} \\ $$

Question Number 87686 Answers: 3 Comments: 0

∫(√((ln(x+(√(1+x^2 ))))/(1+x^2 ))) dx

$$\int\sqrt{\frac{{ln}\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)}{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx} \\ $$

Question Number 87669 Answers: 1 Comments: 4

∫_2 ^( e) ((1/(ln x))−(1/(ln^2 x))) dx?

$$\int_{\mathrm{2}} ^{\:\:\mathrm{e}} \left(\frac{\mathrm{1}}{\mathrm{ln}\:\mathrm{x}}−\frac{\mathrm{1}}{\mathrm{ln}^{\mathrm{2}} \mathrm{x}}\right)\:\mathrm{dx}? \\ $$

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