Question
Download Solution PDFWhat is the value of ∫ ex(sin x - cos x) dx?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
-
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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