What will be the value of this integral? Please explain with proper steps!
Sanjiban Sengupta , 8 Years ago
Grade 12
Integral Calculus> What will be the value of this integral? ...
1 Answers
Nandana
Last Activity: 8 Years ago
hi !
Ans : 5/4√ π erfi(sinx)-1/2 sinx esin2 x
where erfi(sinx) is imaginary error in integration , which is ∫2/ √π eu2 du
EXP :
∫cosx e^sin2 x (1+cos2 x) dx
sinx = y
cosx dx = dy
= ∫ e^y2 (2- y2 ) dy
= 2∫ e^y2 dy -∫ y2 e^y2 dy
2∫ e^y2 dy = √π ∫ 2/√π e^y2 dy
= √π erfi (sinx)
∫ y2 e^y2 dy by using integration by part method
u =y & dv = ye^y2 dy
du = dy v= ½ e^y2
uv -∫v du = ½ y e^y2 - ½∫ e^y2 dy
= ½ sinx e^sin2 x - √π/4 ∫ e^y2 dy
=½ sinx e^sin2 x - √π/4erfi(sinx)
now, the overall integration is -
2∫ e^y2 dy -∫ y2 e^y2 dy = √π erfi (sinx) -[½ sinx e^sin2 x - √π/4erfi(sinx)]
=> 5√π /4 erfi(sinx) - ½ sinx e^sin2 x + K
where , K is orbitary constant
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