Mathematics

# Integrate $\int _{ -\pi /2 }^{ \pi /2 }{ { e }^{ \sin ^{ -2 }{ x } } } .\sin ^{ 2n+1 }{ x } dx=$

##### SOLUTION
Let  $I = \int\limits_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {{e^{{{\left( {{{\sin }^{ - 1}}x} \right)}^2}}}{{\sin }^{2n + 1}}xdx}$
since  $f\left( x \right) = {e^{{{\left( {{{\sin }^{ - 1}}x} \right)}^2}}}{\sin ^{2n + 1}}xdx$ is an odd function
So, $I = 0$

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

#### Realted Questions

Q1 Subjective Hard
$\displaystyle\int \sqrt{\dfrac{\cos^3x}{\sin^{11}x}}dx$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Assertion & Reason Hard
##### ASSERTION

If $n> 1$ then $\displaystyle \int_{0}^{\infty }\frac{dx}{1+x^{n}}=\int_{0}^{1}\frac{dx}{\left ( 1-x^{n} \right )^{1/n}}$

##### REASON

$\displaystyle \int_{a}^{b}f\left ( x \right )dx=\displaystyle \int_{a}^{b}f\left ( a+b-x \right )dx$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Assertion is correct but Reason is incorrect
• C. Both Assertion and Reason are incorrect
• D. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\dfrac {(1)}{(2-3x)^4} dx=$ ?
• A. $\dfrac {(1)}{(2-3x)^4}+C$
• B. $\dfrac {(1)}{-12(2-3x)^3}+C$
• C. none of these
• D. $\dfrac {(1)}{9(2-3x)^3}+C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Evaluate:
$\int { \cfrac { 2x }{ ({ x }^{ 2 }+1)({ x }^{ 2 }+3) } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
If $\dfrac{d}{dx}f(x)=g(x)$ then $\displaystyle \int_{a}^{b}f(x)g(x)dx=$
• A. $\dfrac{f(b)-f(a)}{2}$
• B. $\dfrac{f(a)-f(b)}{2}$
• C. $\dfrac{{f}^{2}(a)-{f}^{2}(b)}{2}$
• D. $\dfrac{{f}^{2}(b)-{f}^{2}(a)}{2}$