JEE Main 2014 · 19 Apr · Shift 1 · Mathematics Q74

← JEE Main 2014 · 19 Apr · Shift 1

MathematicsSingle correct+4 / -1Question 74 of 90

If $m$ is a non - zero number and
$\int\frac{x^{5m-1}+2x^{4m-1}}{(x^{2m}+x^m+1)^3}dx=f(x)+c,$
then $f(x)$ is :

\dfrac{x^{5m}}{2m(x^{2m}+x^m+1)^2}

\dfrac{x^{4m}}{2m(x^{2m}+x^m+1)^2}

\dfrac{2m(x^{5m}+x^{4m})}{(x^{2m}+x^m+1)^2}

\dfrac{(x^{5m}-x^{4m})}{2m(x^{2m}+x^m+1)^2}

Answer and worked solution

Official answer: B

Divide the numerator and denominator by $x^{6m}$ :
$\int\frac{x^{-m-1}+2x^{-2m-1}}{(1+x^{-m}+x^{-2m})^3}dx.$
With $u=1+x^{-m}+x^{-2m}$ , $du=-m(x^{-m-1}+2x^{-2m-1})dx$ :
$-\frac1m\int\frac{du}{u^3}=\frac1{2mu^2}=\frac{x^{4m}}{2m(x^{2m}+x^m+1)^2}.$

Topics:Substitution

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JEE Main 2014 · 19 Apr · Shift 1, Mathematics question 74