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Definition of Functions

If ∫ x 26 x -...

Question

If $\int x^{26} (x - 1)^{17} (5 x - 3) d x = \frac{x^{27} . (x - 1)^{18}}{k} + c,$ where c is the constant of integration, then the value of k is

A

3

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B

4

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C

9

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D

10

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Solution

The correct option is C
9

Differentiating both sides gives

$x^{26} . (x - 1)^{17} (5 x - 3) = \frac{1}{k} [x^{27} . 18 (x - 1)^{17} + (x - 1)^{18} 27 x^{26}]$

$= \frac{x^{26} (x - 1)^{17}}{k} [18 x + 27 (x - 1)]$

$= \frac{x^{26} (x - 1)^{17}}{k} (45 x - 27)$

$= 9 \frac{x^{26} (x - 1)^{17}}{k} (5 x - 3)$

$\Rightarrow k = 9$

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