Mathematics

# $\int _{ -4 }^{ -5 }{ { e }^{ { \left( x+5 \right) }^{ 2 } } } dx+\int _{ 1/3 }^{ 2/3 }{ { e }^{ { 9\left( x-\cfrac { 2 }{ 3 } \right) }^{ 2 } } } dx$ is equal to

$0$

##### SOLUTION
We have given from the question
$\begin{matrix} \int _{ -4 }^{ -5 }{ { e^{ { { \left( { x+5 } \right) }^{ 2 } } } }dx }+\int _{ \frac { 1 }{ 3 } }^{ \frac { 2 }{ 3 } }{ { e^{ 9{ { \left( { x-\frac { 2 }{ 3 } } \right) }^{ 2 } } } }dx } \\ Let \\ x+5=t \\ Now, \\ \int _{ 1 }^{ 0 }{ { e^{ { t^{ 2 } } } }dx }\, \, \, +3\int _{ \frac { 1 }{ 3 } }^{ \frac { 2 }{ 3 } }{ { e^{ { { \left( { 3x-2 } \right) }^{ 2 } } } }dx } \\ Again \\ Let\, \, 3x-2=v \\ \therefore \frac { { dv } }{ { dx } } =3 \\ Now \\ \int _{ 1 }^{ 0 }{ { e^{ { t^{ 2 } } } }dx }\, \, \, +\frac { 1 }{ 3 } \times 3\int _{ -1 }^{ 0 }{ { e^{ { v^{ 2 } } } }dv } \\ \int _{ 1 }^{ 0 }{ { e^{ { t^{ 2 } } } }dx }\, \, \, +\int _{ -1 }^{ 0 }{ { e^{ { v^{ 2 } } } }dv }\, \, \, \, \, \, \, \, \, \, \, \left[ { u=-x } \right] \\ \int _{ 1 }^{ 0 }{ { e^{ { t^{ 2 } } } }dx }\, \, \, +\int _{ 1 }^{ 0 }{ { e^{ { v^{ 2 } } } }dv }\, \, \, \, =0\, \, \\ \end{matrix}$

Hence, the option $(C)$ is the correct answer.

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
Evaluate $\displaystyle \int \frac{\log\left ( x+\sqrt{1+x^{2}} \right )}{\sqrt{1+x^{2}}}dx.$

• A. $\displaystyle \frac{1}{2\left [ \log\left ( x+\sqrt{1+x^{2}} \right ) \right ]^{2}} +c$
• B. $\displaystyle \frac{\left[\log\left ( x+\sqrt{1+x^{2}} \right )\right]^2}{4}+c$
• C. $\displaystyle \frac{\left[\log\left ( x-\sqrt{1+x^{2}} \right )\right]^2}{2}+c$
• D. $\displaystyle \frac{\left[\log\left ( x+\sqrt{1+x^{2}} \right )\right]^2}{2}+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 One Word Medium
Evaluate $\displaystyle \int_\limits{0}^{\pi} \frac{1}{1+ \sin x}dx - 2\int_\limits{0}^{\frac{\pi}{2}} \frac{1}{1+ \sin x}dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate the following integrals:
$\displaystyle \int { \cfrac { a{ x }^{ 2 }+bx+c }{ (x-a)(x-b)(x-c) } } dx$ where $a,b,c$ are distinct.

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
If $a> 0$ and $\displaystyle \int_{0}^{a}\left [ f\left ( x \right )+f\left ( -x \right ) \right ]\: dx= \displaystyle \int_{-a}^{a}\: \phi \left ( x \right )\: dx$ then one of the possible values of $\phi \left ( x \right )$ can be
• A. $-f\left ( x \right )$
• B. $\displaystyle \frac{1}{2}f\left ( x \right )$
• C. none of these
• D. $f\left ( -x \right )$

Find $\displaystyle \int \sqrt{10 - 4x + 4x^2} dx$