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Pattern – 6 : Indefinite Integration

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⋆ Pattern – 6

Method of substitution

$1)\ \ \int\frac{x^2}{\sqrt{x^6-a^6}}\ dx$

Let, $\ x^3=t$

$\therefore\ \ 3x^2\ dx=dt$

$\therefore\ \ x^2dx=\frac{dt}{3}$

$\therefore\ \ I=\frac{1}{3}\ \int\frac{1}{\sqrt{t^2-\left(a^3\right)^2}}\ dt$

$=\frac{1}{3}\times\frac{1}{2a^3}\ \log{\left|\frac{t-a^3}{t+a^3}\right|}+c$

$\int\frac{1}{\sqrt{x^2-a^2}}\ \ dx$

$=\frac{1}{2a}\log{\left|\frac{x-a}{x+a}\right|}+c$

$\therefore\ \ I=\ \frac{1}{6a^3}\ \log{\left|\frac{x^3-a^3}{x^3+a^3}\right|}+c$

$\int\frac{1}{\sqrt x+\ \sqrt[3]{x}}\ dx\$

Let, x = t6