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 February 26th, 2012, 09:37 AM #1 Newbie   Joined: Feb 2012 Posts: 1 Thanks: 0 integration problem from mit who solve Hi guys , this integration problem was in mit integration sheet , so can anyone find the answer? ?. (2x^3 + 5x + 1)e^(2x) dx
 February 26th, 2012, 09:50 AM #2 Senior Member   Joined: Jul 2011 Posts: 227 Thanks: 0 Re: integration problem from mit who solve $\int (2x^3+5x+1)e^{2x}dx$ $=2\int x^3e^{2x}dx + 5 \int xe^{2x}dx + \int e^{2x}dx$ For the first two integrals use integration by parts. For the third one you can use the substitution $2x=t \Rightarrow 2dx=dt$.
 February 27th, 2012, 01:09 PM #3 Senior Member   Joined: May 2011 Posts: 501 Thanks: 5 Re: integration problem from mit who solve Here is a method for integrals of the form $p(x)e^{kx}$, where p(x) is a polynomial of degree n. using $\frac{d}{dx}p(x)e^{2x}$ $\int (2x^{3}+5x+1)e^{2x}dx$ $p(x)=ax^{3}+bx^{2}+cx+d$ then $\frac{d}{dx}(ax^{3}+bx^{2}+cx+d)e^{2x}=(2x^{2}+5x+ 1)e^{2x}$ Use product rule: $2(ax^{3}+bx^{2}+cx+d)e^{2x}+(3ax^{2}+2bx+c)e^{2x}= (2x^{3}+5x+1)e^{2x}$ Expand and equate coefficients: $2a=2$ $3a+2b=0$ $2b+2c=5$ $c+2d=1$ $a=1, \;\ b=-3/2. \;\ c=4, \;\ d=-3/2$ $\frac{d}{dx}(x^{3}-\frac{3}{2}x^{2}+4x-\frac{3}{2})e^{2x}=(2x^{3}+5x+1)e^{2x}$ Integrate both sides: $(x^{3}-\frac{3}{2}x^{2}+4x-\frac{3}{2})e^{2x}=\int (2x^{3}+5x+1)e^{2x}dx$

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