Question

The value of int (x111 + sin111x) dx is: {limit -pi/2 to pi/2}
1.
A. 0
B. pi
C. 112
D. 111​

Answer 1

The value of int (x111 + sin111x) dx is: {limit -pi/2 to pi/2}

1.

A. 0

B. pi

C. 112

D. 111

Answer 2

SOLUTION

TO CHOOSE THE CORRECT OPTION

$\displaystyle \sf{\int\limits_{ - \frac{\pi}{2}}^{ \frac{\pi}{2} } \: ( \: {x}^{111} + { \sin}^{111}x \: )\: \, dx }$

A. 0

$\displaystyle \sf{B. \: \: \pi}$

C. 112

D. 111

CONCEPT TO BE IMPLEMENTED

1. f(x) is said to be odd function if

f(- x) = - f(x)

2. If f(x) is a odd function then

$\displaystyle \sf{\int\limits_{ - a}^{a } \: f( x )\: \, dx = 0}$

EVALUATION

We have to find

$\displaystyle \sf{\int\limits_{ - \frac{\pi}{2}}^{ \frac{\pi}{2} } \: ( \: {x}^{111} + { \sin}^{111}x \: )\: \, dx }$

Let

$\displaystyle \sf{f(x) = {x}^{111} + { \sin}^{111}x \: }$

Now

$\displaystyle \sf{f( - x) = {( - x)}^{111} + \bigg( { \sin \: ( - x)} \bigg)^{111} \: }$

$\displaystyle \implies \: \sf{f( - x) = - {x}^{111} - { \sin}^{111}x \: }$

$\displaystyle \implies \: \sf{f( - x) = - ({x}^{111} + { \sin}^{111}x) \: }$

$\displaystyle \implies \: \sf{f( - x) = - f(x) }$

So f(x) is an odd function

So by the above mentioned formula

$\displaystyle \sf{\int\limits_{ - \frac{\pi}{2}}^{ \frac{\pi}{2} } \: ( \: {x}^{111} + { \sin}^{111}x \: )\: \, dx } = 0$

FINAL ANSWER

Hence the correct option is A. 0

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