int frac{x^{49} tan ^{-1}left(x^{50}right)}{1 | vedclass

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Question

(C) Let $I = \int \frac{x^{49} 	an ^{-1}\left(x^{50}ight)}{1+x^{100}} d x$.Substitute $t = x^{50}$,then $dt = 50x^{49} dx$,which implies $x^{49} dx

Vedclass · Accepted Answer

$\frac{1}{100}$

Vedclass · Answer

(C) Let $I = \int \frac{x^{49} 	an ^{-1}\left(x^{50}ight)}{1+x^{100}} d x$.Substitute $t = x^{50}$,then $dt = 50x^{49} dx$,which implies $x^{49} dx = \frac{1}{50} dt$.Substituting these into the integral,we get:$I = \frac{1}{50} \int \frac{	an ^{-1} t}{1+t^2} dt$.Now,let $u = 	an ^{-1} t$,then $du = \frac{1}{1+t^2} dt$.The integral becomes:$I = \frac{1}{50} \int u du = \frac{1}{50} \cdot \frac{u^2}{2} + c = \frac{u^2}{100} + c$.Substituting back $u = 	an ^{-1} (x^{50})$,we have:$I = \frac{(	an ^{-1} (x^{50}))^2}{100} + c$.Comparing this with the given expression $k(	an ^{-1} (x^{50}))^2 + c$,we find $k = \frac{1}{100}$.

$\int \frac{x^{49} \tan ^{-1}\left(x^{50}\right)}{1+x^{100}} d x=k\left(\tan ^{-1}\left(x^{50}\right)\right)^2+c$,then $k$ is equal to

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