1.

The value of int_(0)^(pi//2) (cos3x+1)/(2cos x-1)dx is

Answer»

2
1
`(1)/(2)`
0

Solution :Let `I=underset(0)OVERSET(pi//2)INT (cos 3x+1)/(2cos x-1)DX`
`=underset(0)overset(pi//2)int (cos 3x-cos""(3pi)/(4))/(2(cosx-cos""(pi)/(3)))dx`
`=underset(0)overset(pi//2)int ((4cos^(3)x-3cosx)-(4cos^(3)""(pi)/(4)-3cos""(pi)/(3)))/(2(cos x-cos""(pi)/(3)))dx`
`=2underset(0)overset(pi//2)int ((cos^(3)x-cos^(3)""(pi)/(3))/(cos x-cos""(pi)/(3)))dx`
`=2underset(0)overset(pi//2)int (cos^(2)x+cos^(2)""(pi)/(3)+cos x cos ""(pi)/(3))dx-(3)/(2)underset(0)overset(pi//2)int 1dx`
`=underset(0)overset(pi//2)int (1+cos2x+(1)/(2)+cosx)dx-(3pi)/(4)`
`=(3pi)/(4)+1-(3pi)/(4)=1`


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