Question
Download Solution PDF∫ ex(sin x - cos x) dx का मान क्या है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
-
खंडशः समाकलन:
∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ [f'(x) ∫ g(x) dx] dx.
- ∫ sin x dx = - cos x + C
गणना:
माना कि I = ∫ ex(sin x - cos x) dx है।
⇒ 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
जैसा कि हम जानते हैं, ∫ ex [f(x) + f'(x)]dx = ex f(x) + c
माना f(x) = -cos x
इसलिए, f'(x) = sin x
अब , 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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