 Hello and welcome to the session. Let us discuss the following question. It says integrate the following function The given function is 1 upon 1 plus cot x. Let us now proceed on with the solution I be the integral one upon 1 plus cot x d x now we know that Caught x is cos x upon sine x. So the integral becomes 1 upon 1 plus Cos x upon sine x d x Which is again equal to integral Sine x upon Sine x plus cos x d x Now multiplying and dividing by 2 we have 1 by 2 into 2 sine x Upon sine x plus cos x d x Now this can be further written as sine x plus cos x plus sine x minus cos x We are adding in subtracting cos x and we are writing 2 sine x is sine x plus sine x This can be further written as 1 by 2 Integral sine x plus cos x upon sine x plus cos x d x plus 1 by 2 integral sine x minus cos x upon sine x plus cos x d x Now this is equal to 1 by 2 integral d x because sine x plus cos x gets cancelled with sine x plus cos x plus 1 by 2 integral sine x minus Cos x upon sine x plus cos x d x Now here we see that the derivative of sine x plus cos x is cos x minus sine x So here we need to have minus sine x and plus cos x So we take negative sine common So this becomes integral d x Taking minus common this becomes minus 1 by 2 integral cos x minus sine x upon sine x plus cos x d x Now equal to sine x plus cos x So d t by d x is equal to cos x minus sine x and this implies d t equal to cos x minus sine x d x So cos x minus sine x is d t by d x so that d t is equal to cos x minus sine x d x So the first integral is equal to 1 by 2 d x minus 1 by 2 integral d t upon t Because cos x minus sine x d x is d t and sine x plus cos x is t now Minus 1 by 2 integral of 1 by t d t is log mod t Plus c where c is the constant of both the integration here So this is equal to x by 2 minus 1 by 2 log mod t is sine x plus cos x So let's substitute It's sine x plus cos x plus c Hence the integral of the given function is x by 2 minus 1 by 2 log mod cos x plus sine x plus c and This completes the question. Bye for now take care. Have a good day