Mathematics

# $\displaystyle \int _{ -\dfrac { 3\pi }{ 2 } }^{- \dfrac { \pi }{ 2 } }{ \left[ { \left( x+\pi \right) }^{ 3 }+\cos ^{ 2 }{ \left( x+3\pi \right) } \right] dx }$ is equal to

##### ANSWER

$\left(\dfrac {\pi}{2} \right)$

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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 Hard
Evaluate $\displaystyle {\int \sin^{-1}\, \sqrt {\frac {x}{a\, +\, x}} dx}$
• A. $(a + x) \tan^{-1} \displaystyle \sqrt \frac {x}{a}\, -\, \sqrt {ax}\, +\, c$
• B. $a \tan^{-1} \displaystyle \sqrt \frac {x}{a}\, -\, \sqrt {ax}\, +\, c$
• C. $(a + x) \tan^{-1} \displaystyle \sqrt {x}\, -\, \sqrt {ax}\, +\, c$
• D. $\displaystyle \left(\frac { ax+1 }{ 2a } \right)\sin^{ -1 }\, \sqrt { \frac { x }{ a\, +\, x } } -\frac { 1 }{ 2 } \sqrt { \frac { x }{ a } } +c$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Let the equation of a curve passing through the point $\displaystyle \left ( 0, 1 \right )$ be given by $\displaystyle y=\int x^{2}.e^{x^{3}}dx.$ If the equation of the curve is written in the form $\displaystyle x=f\left ( y \right )$ then $\displaystyle f\left ( y \right )$ is
• A. $\displaystyle \sqrt{\log _{e}\left ( 3y-2 \right )}$
• B. $\displaystyle \sqrt{\log _{e}\left ( 2-3y \right )}$
• C. none of these
• D. $\displaystyle \sqrt{\log _{e}\left ( 3y-2 \right )}$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate :
$\int {\dfrac{x}{{{{\left( {x\, - 1} \right)}^2}\left( {x + \,2} \right)}}} \,dx$.

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \int _{ 0 }^{ x }{ \cfrac { \sin { x } }{ 1+\cos ^{ 2 }{ x } } } dx=\pi \cfrac { \cos { \alpha } }{ 1-\sin ^{ 2 }{ \alpha } }$
• A. for no value of $\alpha$
• B. for exactly two values of $\alpha$ in $\left( 0,\pi \right)$
• C. for atleast one $\alpha$ in $\left( \pi /2,\pi \right)$
• D. for exactly one $\alpha$ in $\left( 0,\pi /2 \right)$

Asked in: Mathematics - Integrals

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.

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020