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Evaluate : $ \int\large\frac{x^2}{1+x^3}$$dx $

1 Answer

  • Method of substitution:
  • Given $\int f(x)dx$ can be transformed into another form by changing independent variable x to t by substituting x=g(t).
  • Consider $I=\int f(x)dx.$
Step 1:
Let $1+x^3=t$
On differentiating with respect to $x$
$I=\int \large\frac{dt/3}{1+x^3}$
$\;\;=\large\frac{1}{3}\int \large\frac{dt}{t}$
Step 2:
On integrating we get,
$\large\frac{1}{3}$$\log \mid t\mid+c$
Substituting for $t$ we get,
$I=\large\frac{1}{3}$$\log \mid 1+x^3\mid +c$
answered Sep 26, 2013 by sreemathi.v