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

$e^5$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

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
$\dfrac{I_1}{I_2}$ is
• A. $2$
• B. $1$
• C. $-\displaystyle \frac{1}{2}$
• D. $\displaystyle \frac{1}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
$\int \sqrt[3]{x} dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate $\int \dfrac {1}{\sqrt {7-6x-{x}^{2}}}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Write the value of $\displaystyle \int Xa^{x^{2}+1}dx$

1 Verified Answer | Published on 17th 09, 2020

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
$\displaystyle \int u\left ( x \right )v\left ( x \right )dx=u\left ( x \right )v_{1}-u'\left ( x \right )v_{2}\left ( x \right )+u''\left ( x \right )v_{3}\left ( x \right )+...+\left ( -1 \right )^{n-1}u^{n-1}\left ( x \right )V_{n}\left ( x \right ) \\ -\left ( -1 \right )^{n-1}\int u^{n}\left ( x \right )V_{n}\left ( x \right )dx$
where  $\displaystyle v_{1}\left ( x \right )=\int v\left ( x \right )dx,v_{2}\left ( x \right )=\int v_{1}\left ( x \right )dx ..., v_{n}\left ( x \right )= \int v_{n-1}\left ( x \right )dx$
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.