PLEASE SOLVE AND GIVE CORRECT ANSWER Question70of90∫ 0xsint dt where integer x ∈(2nπ,4n + 1π, n ∈ - Maths - Integrals

Question

PLEASE SOLVE AND GIVE CORRECT ANSWER

Q u e s t i o n 70 o f 90 ∫  0 x sin t   d t  
w h e r e   i n t e g e r   x   ∈ ( 2 n π
, 4 n   +   1 π ,   n   ∈   N
  and     d e n o t e s   t h e   g r e a
t e s t   i n t e g e r   f u n c t i o n   i s
  e q u a l   t o 1   - n π 2   - n  
+   1 π 3   - 2 nπ 4   - 2 n   +
  1 π

Ghanshyam Dhakar · Accepted Answer

Dear student,
let I = ∫0xsinxdxwhen x  =2nπthen I = ∫02nπsinxdxuse  ∫0n×Tfxdx = n∫0Tfxdx T is time period of fxI = n∫02πsinxdxI = n∫0πsinxdx+n∫π2πsinxdxwhen x∈0,π then sinx∈0,1  and when x∈π,2π then sinx∈-1,0I = n∫0π0×dx+n∫π2π-1dxI = -nxπ2π = -nπwhen x  =4n+1πthen I = ∫04n+1πsinxdxI = ∫04nπsinxdx+∫4nπ4nπ+πsinxdxuse  ∫0n×Tfxdx = n∫0Tfxdx T is time period of fxI = 2n∫02πsinxdx+∫4nπ4nπ+πsinxdxI = 2n∫0πsinxdx+2n∫π2πsinxdx+∫4nπ4nπ+πsinxdxwhen x∈0,π then sinx∈0,1  and when x∈π,2π then sinx∈-1,0I = 2n∫0π0×dx+2n∫π2π-1dx+∫4nπ4nπ+π0×dxI = -2nxπ2π = -2nπ   answer
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PLEASE SOLVE AND GIVE CORRECT ANSWER Q u e s t i o n 70 o f 90 ∫ 0 x sin t d t w h e r e i n t e g e r x ∈ ( 2 n π , 4 n + 1 π , n ∈ N and . d e n o t e s t h e g r e a t e s t i n t e g e r f u n c t i o n i s e q u a l t o . 1 . - n π 2 . - n + 1 π 3 . - 2 nπ 4 . - 2 n + 1 π