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  4. If curve f(x)=∫ x11(2 x3+3 x+4)3(1+x+x3) d x passes through origin f(1)=(p/q) where p q are relatively prime then p-q is

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Q. If curve $f(x)=\int x^{11}\left(2 x^{3}+3 x+4\right)^{3}\left(1+x+x^{3}\right) d x$ passes through origin & $f(1)=\frac{p}{q}$ where $p \& q$ are relatively prime then $p-q$ is

Integrals

Solution:

$f(x)=\int x^{11}\left(2 x^{3}+3 x+4\right)^{3}\left(1+x+x^{3}\right) d x$
$2 x^{6}+3 x^{4}+4 x^{3}=t$
$f(x)=\frac{1}{48}\left(2 x^{6}+3 x^{4}+4 x^{3}\right)^{4}$
$f(1)=\frac{9^{4}}{48}=\frac{2187}{16}=\frac{p}{q}$
$\therefore p-q=2171$