If _0^2x(2-x)^b ,dx= 32 7 , where b N then b=

MHT CET 2026 Mathematics question

If , where then

(A)5
(B)6Correct
(C)7
(D)8

Answer: (B)

Solution

Use the property \int_0^af(x)dx=\int_0^af(a-x)dx with : \int_0^2x(2-x)^bdx=\int_0^2(2-x)x^bdx=2\int_0^2x^bdx-\int_0^2x^{b+1}dx=\frac{2\cdot2^{b+1}}{b+1}-\frac{2^{b+2}}{b+2}=2^{b+2}\left[\frac{1}{b+1}-\frac{1}{b+2}\right]=\frac{2^{b+2}}{(b+1)(b+2)}. Set

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Question ID VK-100347