 Hello and welcome to the session. Let us discuss the following question. Question says, choose the correct answer, the value of definite integral from minus pi upon 2 to pi upon 2 x cube plus x cos x plus tan x raised to the power 5 plus 1 dx is a 0 b 2 c pi d 1. We have to choose the correct answer from a, b, c and d. First of all, let us understand that definite integral from minus a to a f x dx is equal to 2 multiplied by definite integral 0 to a f x dx if f is an even function. That is, if f minus x is equal to f x, then f x is an even function. Also, definite integral from minus a to a f x dx is equal to 0 if f is an odd function. That is, if f minus x is equal to minus f x. Now, this property of the definite integral is the key idea to solve the given question. Let us now start with the solution. Now, we have to find the value of definite integral minus pi upon 2 to pi upon 2 x cube plus x cos x plus tan x raised to the power 5 plus 1 dx. Now, this integral is further equal to definite integral from minus pi upon 2 to pi upon 2 x cube dx plus definite integral from minus pi upon 2 to pi upon 2 x cos x dx plus definite integral from minus pi upon 2 to pi upon 2 tan x raised to the power 5 dx plus definite integral from minus pi upon 2 to pi upon 2 dx. Now, this integral is equal to 0. Clearly, we can see x cube is an odd function where f x is equal to x cube f minus x is equal to minus x cube. So, we get f minus x is equal to minus f x. So, x cube is an odd function. Now, using the property given in key idea, we get this integral is equal to 0. Similarly, we can see x cos x is an odd function. If f x is equal to x cos x, f minus x is equal to minus x cos x. Now, we get f x is equal to minus f x. So, x cos x is an odd function. Now, using the property given in key idea, we get this integral is equal to 0. Now, let us consider tan x raised to the power 5. Now, clearly we can see if f x is equal to tan x raised to the power 5, then f minus x is equal to minus tan x raised to the power 5. So, we get f x is equal to minus f x here. So, this implies tan x raised to the power 5 is an odd function. Now, since tan x raised to the power 5 is an odd function, this integral is equal to 0. Using the property given in key idea, we get this integral is equal to 0. Now, let us consider this integral. Now, we can write this integral as definite integral from minus pi upon 2 to pi upon 2 1 dx. Now, this is a constant function. If f x is equal to 1, then f minus x is also equal to 1. We get f x is equal to f minus x equal to 1. Now, we get 1 is an even function. Now, using the property given in key idea, we can write this integral as 2 multiplied by definite integral from 0 to pi upon 2 dx. Now, this is further equal to 2 multiplied by x where limits are 0 to pi upon 2. We know integral of 1 with respect to x is x. Now, we will substitute upper and lower limit in this function. We get 2 multiplied by pi upon 2 minus 0. Now, simplifying further, we get 2 multiplied by pi upon 2. Now, 2 and 2 will get cancelled and we get pi. So, we get given definite integral is equal to pi. So, our correct answer is c. So, this is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.