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Evaluate the following integral ∫1/(cos⁡x (〖sin〗^2 x+1) ) dx
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Evaluate the following integral ∫1/(cos⁡x (〖sin〗^2 x+1) ) dx

hamza
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2022-05-18
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Evaluate the following integral ∫1/(cos⁡x (〖sin〗^2 x+1)  )  dx

Mathematics,
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hamza
2022-05-18
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Put  z=sin⁡x     ,    cos⁡z=√(1-z^2 )    ,   dx=1/(√(1-z^2 )  ) dz
I=∫〖1/(√(1-z^2 )  (z^2+1) )  dz/(√(1-z^2 )  )〗=∫dz/(1-z^2 )(1+z^2 )
Using partial fractions:
I=1/2 ∫dz/((1-z^2 ) )+1/2 ∫dz/((1+z^2 ) )=1/2  tanh⁡〖x^(-1) 〗+1/2  tan^(-1)⁡z+c

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