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Pattern - 43(i) of Indefinite Integration

February 24, 2021
Category: Pattern - 43(i) of Indefinite Integration

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⋆ Pattern – 43(i)

Integrals of differential binomials and chebyslev’s criterion

∫ $x^m\left(a+bx^n\right)^p\ dx$ : p ≠ 0

(i) If p ∈ n, then expand $\left(a+bx^n\right)^p$ by binomial theorem & integrate.

1) Evaluate : $\int x^\frac{-2}{3}\ \left(5+4x^2\right)^3dx$

Solution : Let,

I= $\ \int x^\frac{-2}{3}\ \left(5+4x^2\right)^3dx$

m= $\ \frac{-2}{3},\ n=2\ ,\ p = 3 \in N$

$\therefore\ \ I=\ \int x^{\frac{-2}{3}\ }\left[\left(5\right)^2+3\left(5\right)^2\ \left(4x^2\right)+3\left(5\right)\left(4x^2\right)^2+\ \left(4x^2\right)^3\right]\ dx$

$=\int x^\frac{-2}{3}\ \left(125+300x^2+240\ x^4+64x^6\right)\ dx$

$=\int\left(125\ x^\frac{-2}{3}+300\ x^{\frac{4}{3}\ }+240\times x^\frac{10}{3}+64\ x^\frac{16}{3}\right)dx$

$=\ \frac{125\times x^\frac{1}{3}}{\frac{1}{3}}+\ \frac{300\ x^\frac{7}{3}}{\frac{7}{3}}+\ \frac{240\times x^{\frac{13}{3}\ }}{\frac{13}{3}}+\ \frac{64x^\frac{19}{3}}{\frac{19}{3}}+c$