 Hi, and welcome to our session. Let us discuss the following question. The question says, by using the properties of definite integrals, evaluate the following. Integral of log 1 plus tan x from 0 to pi by 4. Before solving this question, we should know that integral of fx from 0 to a is equal to integral of fa minus x from 0 to a. The knowledge of this property is the key idea in this question. This is a solution. Let i is equal to integral of log 1 plus tan x from 0 to pi by 4. But the key idea, we know that integral of fx from 0 to a is equal to integral of fa minus x from 0 to a. So by using this property, we get i as integral of log 1 plus 4 minus x as a is equal to pi by 4 from 0 to pi by 4. This is equal to integral of log 1 plus, now using formula of tan a minus tan v, we get tan pi by 4 minus tan x by 1 plus tan pi by 4 into tan x from 0 to pi by 4. This is equal to integral of log 1 plus 1 minus tan x by 1 plus tan x from 0 to pi by 4. And this is equal to integral of log 2 by 1 plus tan x from 0 to pi by 4. Log a by v is equal to log a minus log b. So this is equal to integral of log 2 minus log 1 plus tan x from 0 to pi by 4. Now this is equal to log 2 into integral of 1 from 0 to pi by 4 minus integral of log 1 plus tan x from 0 to pi by 4. Let's name this equation as equation number 1 and this as 2. Now on adding 1 and 2, we get plus i is equal to integral of log 1 plus tan x from 0 to pi by 4 plus log 2 into integral of 1 from 0 to pi by 4 minus integral of log 1 plus tan x from 0 to pi by 4. Now this is equal to log 2 into integral of 1 from 0 to pi by 4. We know that integral of 1 with respect to x is equal to x. So this is equal to log 2 into x where the lower limit is 0 and upper limit is pi by 4. Now we will first substitute the upper limit. By substituting upper limit, we get pi by 4 log 2 minus. Now we will substitute the lower limit at the 0. So we have 0 into log 2. And this is equal to pi by 4 log 2. So 2i is equal to pi by 4 log 2. This implies i is equal to pi by 8 log 2. Hence our required answer is pi by 8 log 2. So this completes the session i and take care.