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  4. If ∫ x23(1-x)31(3-7 x) d x=(xp(1-x)q/r)+C where p, q, r ∈ N and C is integration constant then H.C.F. of (p, q, r) is

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Q. If $\int x^{23}(1-x)^{31}(3-7 x) d x=\frac{x^p(1-x)^q}{r}+C$ where $p, q, r \in N$ and $C$ is integration constant then H.C.F. of $(p, q, r)$ is

Integrals

Solution:

$x^{23}(1-x)^{31}(3-7 x)=x^{23}(1-x)^{31}[3(1-x)-4 x] $
$=3 x^{23}(1-x)^{32}-4 x^{24}(1-x)^{31} $
$=\frac{1}{8} \cdot \frac{d}{d x}\left(x^{24}(1-x)^{32}\right)$