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Question Number 53081 by cesar.marval.larez@gmail.com last updated on 17/Jan/19

	Answered by tanmay.chaudhury50@gmail.com last updated on 17/Jan/19

	$1) \int e^{2 x} s i n x d x$ $e^{2 x} \int s i n x d x - \int [\frac{d (e^{2 x})}{d x} \int s i n x d x] d x$ $= e^{2 x} \times - c o s x - \int 2 e^{2 x} (- c o s x) d x$ $= - e^{2 x} c o s x + 2 \int e^{2 x} c o s x d x$ $= - e^{2 x} c o s x + 2 I_{1}$ $I_{1} = \int e^{2 x} c o s x d x$ $= e^{2 x} \int c o s x d x - \int [\frac{d (e^{2 x})}{d x} \int c o s x d x] d x$ $= e^{2 x} s i n x - \int 2 e^{2 x} s i n x d x$ $= e^{2 x} s i n x - 2 I$ $I = - e^{2 x} c o s x + 2 I_{1}$ $I = - e^{2 x} c o s x + 2 (e^{2 x} s i n x - 2 I)$ $I = - e^{2 x} c o s x + 2 e^{2 x} s i n x - 4 I$ $5 I = - e^{2 x} c o s x + 2 e^{2 x} s i n x$ $I = \frac{1}{5} (- e^{2 x} c o s x + 2 e^{2 x} s i n x)$
	Commented by Otchere Abdullai last updated on 17/Jan/19

	$i n f a c t a m l e a r n i n g a l o t f r o m$ $m a t h e m a t i t i a n s o n t h i s g r e a t$ $p l a t f o r m!$
	Commented by tanmay.chaudhury50@gmail.com last updated on 17/Jan/19

	$t h a n k y o u . . .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 17/Jan/19

	$\int \frac{s i n x}{{c o s}^{3} x} d x$ $= (- 1) \int \frac{d (c o s x)}{{c o s}^{3} x}$ $= (- 1) \times \frac{{(c o s x)}^{- 3 + 1}}{- 3 + 1} + c$ $= \frac{1}{2 {c o s}^{2} x} + c$
	Answered by afachri last updated on 17/Jan/19

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