 Hi and welcome to the session. Let us discuss the following question. The question says integrate the following functions 1 by square root of sine q x into sine x plus alpha Let's now begin with the solution let i is equal to integral of 1 by square root of sine q x Into sine x plus alpha With respect to x now consider sine q x Into sine x plus alpha We know that sine a plus v is equal to sine a cos v plus cos a sine b so using this formula This is equal to sine q x into sine x cos alpha plus Cos x into sine alpha We are going to take common sine x from both these terms So this is equal to sine x to the power 4 into cos alpha plus cot x into sine alpha Now we will put Sine x to the power 4 into cos alpha plus cot x sine alpha in place of sine q x into sine x plus alpha So now i is equal to integral of 1 by square root of Sine x to the power 4 into cos alpha plus cot x sine alpha gx this is equal to integral 1 by sine square x Into square root of cos alpha plus cot x sine alpha d x and This is equal to Integral cosecant square x by Square root of cos alpha plus cot x sine alpha d x let's now put cos alpha plus cot x sine alpha as t Now we will find dt by dx dt by dx is equal to minus cosecant square x sine alpha this implies cosecant square X dx is equal to minus dt by sine alpha so now Putting t in place of cos alpha plus cot x sine alpha and Minus dt by sine alpha in place of cosecant square x dx we get integral minus dt by sine alpha into Square root of t This is equal to minus 1 by sine alpha Into integral t to the power minus 1 by 2 dt We know that Integral x to the power n t x is equal to x to the power n plus 1 by n plus 1 plus c So you're using this formula. This is equal to t to the power Minus 1 by 2 plus 1 by minus 1 by 2 plus 1 plus c Now this is equal to Minus 2 by sine alpha into square root of t plus c we have assumed cos alpha Plus cot x sine alpha as t Now this implies cos alpha plus cos x by sine x Into sine alpha is equal to t this implies cos alpha sine x Plus cos x sine alpha By Sine x is equal to t By the formula of sine a plus v numerator is equal to sine x plus alpha so t is equal to sine x plus alpha by Sine x so now replacing t by sine x plus alpha by sine x we get Minus 2 by sine alpha Into square root of Sine x plus alpha by sine x Plus c Hence a required answer is minus 2 by sine alpha Into square root of sine x plus alpha by sine x Plus c so this completes the session hi and take care