how to integrate 1/(secx + cosecx) please help in integrating these

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Arun · Answer

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.

Regards

Arun (askIITians forum expert)

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how to integrate 1/(secx + cosecx) please help in integrating these

how to integrate 1/(secx + cosecx) please help in integrating these

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