Mathematics

$$\displaystyle \int\limits_{ - 4} {{e^{{{\left( {x + 5} \right)}^2}}}dx\, + 3} \,\,\,\int_{\frac{1}{3}}^{\frac{2}{3}} {{e^{9{{\left( {\frac{{x - 2}}{3}} \right)}^2}}}} $$ is equal to 


ANSWER

$$e^5$$


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Single Correct Medium Published on 17th 09, 2020
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Q1 Single Correct Hard
If $$\displaystyle I_1=\int_{ -100 }^{101}\frac{ \:dx}{(5+2x-2x^2)(1+e^{(2-4x)} )}$$and $$\displaystyle I_2=\int_{ -100 }^{101}\frac{ \:dx}{(5+2x-2x^2)}$$ then
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Q5 Passage Hard
In calculating a number of integrals we had to use the method of integration by parts several times in succession.
The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts
$$\displaystyle \int u\left ( x \right )v\left ( x \right )dx=u\left ( x \right )v_{1}-u'\left ( x \right )v_{2}\left ( x \right )+u''\left ( x \right )v_{3}\left ( x \right )+...+\left ( -1 \right )^{n-1}u^{n-1}\left ( x \right )V_{n}\left ( x \right ) \\ -\left ( -1 \right )^{n-1}\int u^{n}\left ( x \right )V_{n}\left ( x \right )dx$$ 
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Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration by parts is especially useful when  calculating $$\displaystyle \int P_{n}\left ( x \right )Q\left ( x \right )dx,$$ where $$\displaystyle P_{n}\left ( x \right )$$ is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n+1 times.

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