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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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integrate : $\int (x^3-1)^{\large\frac{1}{3}}.x^5$$dx$

$(a)\;x^2+c \qquad(b)\;x^4+x^3+c \qquad(c)\;\frac{1}{7} (x^3-1) ^{7/3} +\frac{1}{4}(x^3-1)^{4/3}+c \qquad (d)\;(x^4-1) ^{7/3} +(x^3-1)^{4/3}+c$

Can you answer this question?
 
 

1 Answer

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$(x^3-1) =t$
differentiate with respect to x
$3x^2 dx=dt$
=>$\int (t) ^{1/3}. (t+1).\large\frac{dt}{3} $
=> $\large\frac{1}{3} $$\int t^{4/3}+ t^{1/3}$$dt$
=> $\large\frac{1}{3 } \times \frac{3}{7} \times t^{\frac{7}{3}}+ \frac{1}{3} \times \frac{9}{4} t^{\frac{4}{3}}+c$
=> $\large\frac{1}{7}$$ (x^3-1)^{7/3}+\frac{1}{4} (x^3-1)^{4/3}+c$
Hence c is the correct answer.

 

answered Dec 24, 2013 by meena.p
 
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