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NCERT Class XII Mathematics Chapter - - Solutions

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Choose the correct answer in Exercises 206 and 207.
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Question : 207 of 271
Marks: +1, -0
If f(x)=∫0xtsin⁡tdtf(x)=\int\limits_{0}^{x} t\sin t dt, then f′(x)f'(x) is
Solution:  
Let f(x)=∫0xtsin⁡tdtf(x)=\int\limits_{0}^{x} t\sin t dt
f(x)=[t(−cos⁡t)−∫1⋅(−cos⁡t)dt]0xf(x)=\left[ t(-\cos t)-\int 1\cdot(-\cos t)dt \right]_0^x
=[−tcos⁡t+sin⁡t]0x=\left[ -t\cos t+\sin t \right]_0^x
f(x)=−xcos⁡x+sin⁡xf(x) = -x\cos x + \sin x
∴f′(x)=−[x(−sin⁡x)+cos⁡x]+cos⁡x\therefore f'(x) = -\left[ x(-\sin x)+\cos x \right] + \cos x =xsin⁡x−cos⁡x+cos⁡x= x\sin x - \cos x + \cos x =xsin⁡x= x\sin x
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