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Pattern - 43(ii) of Indefinite Integration

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⋆ Pattern – 43 (ii)

Integrals of differential binomials and chebyslev’s  criterion

∫dx  : p ≠ 0

(ii) If p is a negative integer and m,n are rational numbers, put x = t^k ; k = LCM of denominator of m & n.

 

2) Evaluate : \int x^\frac{-2}{3}\ \left(1+x^\frac{2}{3}\right)^{-1}\ dx

Solution.  let,

I= \int x^\frac{-2}{3}\left(1+x^\frac{2}{3}\right)^{-1}\ dx

Here, m = \frac{-2}{3}, n = \frac{2}{3},  p = -1

Here, p is -ve integer, m & n are rational nos.

k = LCM of 3 & 3 = 3

Let, x = t^3

∴  dx = 3t^2 dt

\therefore\ \ I=\ \int\left(t^3\right)^\frac{-2}{3}\ \left[1+\ \left(t^3\right)^\frac{2}{3}\right]^{-1}\times3t^2\ dt=3 \int{t^{-2}\ \left(1+t^2\right)^{-1}}\times t^2\ dt

=3\ \int\frac{1}{1+t^2}\ dt

=3\times\frac{1}{1}\tan^{-1}{\frac{t}{1}}+c

=3\times\tan^{-1}{\left(x^\frac{1}{3}\right)}+c

 

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