 Hello and welcome to the session. I am Deepika here. Let's discuss a question which says evaluate the following definite integral integral from 0 to 1 x into e raise to power x plus sin pi x upon 4 dx. Now in this question we will use the second fundamental theorem of integral calculus. So let's start the solution. Let i is equal to integral from 0 to 1 x into e raise to power x plus sin pi x upon 4 dx. Let us first solve the definite integral x into e raise to power x plus sin pi x upon 4 dx and this is equal to integral of x into e raise to power x dx plus integral of sin pi x upon 4 dx. So given integral we will solve by pass that is we will perform integration by pass. So this is equal to let us take x as a first function and e raise to power x as a second function. So first function into integral of the second minus integral of derivative of the first function into integral of the second function integral of sin pi x upon 4 dx. Now this is equal to x into integral of e raise to power x which is e raise to power x only minus integral of derivative of x is 1 into integral of e raise to power x which is e raise to power x. Integral of sin theta is minus cos theta so integral of sin pi x upon 4 is minus is equal to e raise to power x minus e raise to power x minus 4 over pi x upon 4 is equal to that is the antiderivative of the given function x into e raise to power x plus sin pi x upon 4. Therefore fundamental theorem is equal to f1 minus f of 1 is equal to 1 into e raise to power 1 minus e raise to power 1 minus 4 over pi cos pi by 4 minus f0 is 0 into e raise to power x which is 0 minus e raise to power 0 minus 4 over pi cos pi by 4 into 0 which is cos 0. So this is equal to minus 4 over pi into cos pi by 4 which is 1 over root 2 minus 0 minus e raise to power 0 which is 1 cos 0. Now cos 0 is 1 so 4 over pi into 1 is equal to 4 over root 2 is equal to 1 plus 4 over pi so this is equal to 1 plus 4 over pi minus 2 root 2 over pi. Hence the answer for the question is 1 plus 4 over pi minus 2 root 2 over pi. Hope the solution is clear to you. Bye and take care.