how to integrate 1/(secx + cosecx) please help in integrating these
how to integrate 1/(secx + cosecx) please help in integrating these
Mukul anand , 8 Years ago
Grade 12
Integral Calculus> how to integrate 1/(secx + cosecx) please...
1 Answers
Arun
Dear student
I = ∫ [ 1 / ( sec x + csc x ) ] dx
..= ∫ { 1 / [ (1/cos x) + (1/sin x) ] } dx
..= ∫ [ ( cos x + sin x ) / ( sin x · cos x ) ] dx
..= 2 ∫ [ ( cos x + sin x ) / ( 2 sin x. cos x ) ] dx
..= 2 ∫ { ( cos x + sin x ) / [ 1 - ( 1 - 2 sin x. cos x ) ] } dx ...★★★
..= 2 ∫ { ( cos x + sin x ) / [ 1 - ( sin² x + cos² x - 2 sin x. cos x ) ] } dx ... ★★★
..= 2 ∫ { ( cos x + sin x ) / [ 1 - ( sin x - cos x )² ] } dx
..= 2 ∫ [ 1 ( 1 - u² ) ] du, ......... where ... u = sin x - cos x
..= ∫ { [ 1 / (1+u) ] + [ 1 / (1-u) ] } du ..... Partial Fractions
..= ln | 1+u | - ln | 1-u | + C
..= ln | (1+u) / (1-u) | + C
..= ln | ( 1 + sin x - cos x ) / ( 1 - sin x + cos x ) | + C ............... Ans.
..= ∫ { 1 / [ (1/cos x) + (1/sin x) ] } dx
..= ∫ [ ( cos x + sin x ) / ( sin x · cos x ) ] dx
..= 2 ∫ [ ( cos x + sin x ) / ( 2 sin x. cos x ) ] dx
..= 2 ∫ { ( cos x + sin x ) / [ 1 - ( 1 - 2 sin x. cos x ) ] } dx ...★★★
..= 2 ∫ { ( cos x + sin x ) / [ 1 - ( sin² x + cos² x - 2 sin x. cos x ) ] } dx ... ★★★
..= 2 ∫ { ( cos x + sin x ) / [ 1 - ( sin x - cos x )² ] } dx
..= 2 ∫ [ 1 ( 1 - u² ) ] du, ......... where ... u = sin x - cos x
..= ∫ { [ 1 / (1+u) ] + [ 1 / (1-u) ] } du ..... Partial Fractions
..= ln | 1+u | - ln | 1-u | + C
..= ln | (1+u) / (1-u) | + C
..= ln | ( 1 + sin x - cos x ) / ( 1 - sin x + cos x ) | + C ............... Ans.
Regards
Arun (askIITians forum expert)
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