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Home  >>  CBSE XII  >>  Math  >>  Integrals
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Integrate the function\[\int\frac{x^3}{\sqrt{1-x^8}}\]

$\begin{array}{1 1} \large \frac{1}{4} \sin^{-1}(x^4)+c \\ \large \frac{1}{4} \cos^{-1}(x^4)+c \\ \large \frac{1}{4} \sin(x^4)+c \\ \large \frac{1}{4} \cos (x^4)+c\end{array} $

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1 Answer

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Toolbox:
  • (i) If in a function $\int f(x)dx,\;let\;f(x)=t,\;then\;f'(x)=dt,\;then \;\int f(x)dx=\int tdt$
  • (ii)$\int \large\frac{dx}{\sqrt {a^2-x^2}}=\sin^{-1}(x/a)+c$
given $I=\large\int\frac{x^3}{\sqrt{1-x^8}}dx$
 
Let $x^4=t;$ on differentiating we get
 
$4x^3dx=dt$
 
$=>x^3dx=\frac{dt}{4}$
 
Hence $I=\int \frac{dt/4}{\sqrt {1-t^2}}=\frac{1}{4}\int \frac{dt}{\sqrt{1-t^2}}$
 
On integrating we get, since it is of the form $\int \frac{dx}{\sqrt {a^2-x^2}}$
 
$I=\frac{1}{4}\int \frac{dt}{\sqrt{1-t^2}}=\sin^{-1}(t)+c$
 
Substituting for t we get
 
$I=\frac{1}{4} \sin^{-1}(x^4)+c$

 

answered Feb 17, 2013 by meena.p
 
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