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

  1. - ex cos x + C
  2. ex sin x + C
  3. ex sec x + C
  4. None of these.

Answer (Detailed Solution Below)

Option 1 : - ex cos x + C
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Detailed Solution

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Concept:

  • Integration by Parts:

    ∫ f(x) g(x) dx = 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 - ∫ [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

 

As we know, ∫ ex [f(x) + f'(x)]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 [f(x) + f'(x)]dx

=  ex f(x) + c

- ex cos x + C

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