The correct option is A ex(cos(x)+sin(x))2
By seeing two functions multiplied, one of them being ex, we are tempted to use Integration by parts method to solve it. Here also, our approach will be the same.
So, according to ILATE rule, our first function will be cos(x) , and second function will be ex.
Let I=∫ex cos (x) dx
I=cos(x). ex−∫ ex (−sin (x)) dx
Or I=cos(x). ex+∫ ex sin (x) dx
Let I1=∫ ex sin (x) dx
I=cos(x). ex+I1……(1)
We can see that the integral is similar to the integral we are trying to solve. We’ll apply integration by parts methods here as well.
Here, sin(x) will be the first function here, and ex will be the second.
I1=sin(x). ex−∫ex cos (x) dx
Let's substitute I1=sin(x). ex−∫ex cos (x) dx in the 1st equation.
So, I=cos(x).ex+sin(x). ex−∫ ex cos (x)
We can see that ∫ex cos(x) in the above equation is nothing but I.
I=cos(x). ex+sin(x). ex−I
Or 2I=cos(x). ex+sin(x). ex
I=ex(cos(x)+sin(x))2