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NCERT Class XII Mathematics Chapter - - Solutions

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Choose the correct answers in Exercises 269 to 272.
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Question : 271 of 271
Marks: +1, -0
If f(a+b−x)=f(x)f (a + b - x) = f (x), then ∫abxf(x)dx\int\limits_{a}^{b}xf(x)dx is equal to
Solution:  
I=∫abxf(x)dxI=\int\limits_{a}^{b}xf(x)dx
Let a+b−x=za + b - x = z ⇒−dx=dz\Rightarrow - dx = dz
When x=a,z=bx = a, z = b and when x=b,z=ax = b, z = a
I=−∫ba(a+b−z)f(z)dzI=-\int\limits_{b}^{a}(a+b-z)f(z)dz
=∫ab(a+b)f(z)dz−∫abzf(z)dz=\int\limits_{a}^{b}(a+b)f(z)dz-\int\limits_{a}^{b}zf(z)dz
=(a+b)∫abf(x)dx−∫abxf(x)dx=(a+b)\int\limits_{a}^{b}f(x)dx-\int\limits_{a}^{b}xf(x)dx
=(a+b)∫abf(x)dx−I=(a+b)\int\limits_{a}^{b}f(x)dx-I
2I=(a+b)∫abf(x)dx2I=(a+b)\int\limits_{a}^{b}f(x)dx
Hence I=a+b2∫abf(x)dxI=\frac{a+b}{2}\int\limits_{a}^{b}f(x)dx
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