What is the value of ∫ ex(sin x - cos x) dx?

What is the value of ∫ ex(sin x - cos x) dx? Correct Answer - e^x cos x + C

Concept:

• Integration by Parts:

  ∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫  dx.

• ∫ sin x dx = - cos x + C

Calculation:

Let I = ​​∫ ex(sin x - cos x) dx.

⇒ I = ∫ ex sin x dx - ∫ ex cos x dx

⇒ I = ex ∫ sin x dx - ∫ dx - ∫ ex cos x dx

⇒ I = - ex cos x dx + ∫ ex cos x dx - ∫ ex cos x dx + C

⇒ I = - ex cos x + C

dx = ex f(x) + c

Let f(x) = -cos x

So, f'(x) = sin x

Now, I =  ∫ ex(sin x - cos x) dx

=  ​​∫ ex(- cos x + sin x) dx

= ∫ ex dx

=  ex f(x) + c

= - ex cos x + C