 Hi and welcome to the session. Let us discuss the following question, question says, choose the correct answer integral dx upon x square plus 2x plus 2 equals A x tan inverse x plus 1 plus c B tan inverse x plus 1 plus c C x plus 1 multiplied by tan inverse x plus c d tan inverse x plus c. We have to choose the correct answer from A, B, C and D. First of all let us understand that integral dx upon x square plus A square is equal to 1 upon A multiplied by tan inverse x upon A plus c where c is the constant of integration. We will use this formula as our key idea to solve the given question. Let us now start with the solution. We have to find integral dx upon x square plus 2x plus 2. Now let us consider the expression x square plus 2x plus 2. It can be written as x square plus 2x plus 1 plus 1. 2 can be written as 1 plus 1. Now we also know that 1 raised to any power is equal to 1 only. So, we will assign power 2 to both the 1. Now these three terms form x plus 1 whole square and we will write this term as it is. We know A plus B whole square is equal to A square plus B square plus 2ab. Here we have written x plus 1 whole square for A square plus B square plus 2ab. Now we can write integral dx upon x square plus 2x plus 2 is equal to integral dx upon x plus 1 whole square plus 1 square. Now we know derivative of x plus 1 is 1. So, we will put x plus 1 equal to t. Now differentiating both the sides with respect to x we get 1 is equal to dt upon dx. Now this further implies dx is equal to dt. Now we can write integral dx upon x plus 1 whole square plus 1 square is equal to integral of dt upon t square plus 1 square. We know dx is equal to dt and x plus 1 is equal to t. So, this integral is equal to this integral. Now using the formula given in key idea we get this integral is equal to tan inverse t plus c. Comparing this integral with the formula given in key idea clearly we can see here variable x is replaced by variable t and A is equal to 1. So, we get 1 upon A is equal to 1 only tan inverse t. Now we also know that t is equal to x plus 1. So, substituting x plus 1 for t we get tan inverse x plus 1 plus c. So, our required answer is integral dx upon x square plus 2x plus 2 is equal to tan inverse x plus 1 plus c. So, the correct answer is b. So, b is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.