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Indefinite Integration

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⋆ Pattern – 1 Wrong interpretation of basic formula

$\int\frac{1}{x^2+16}dx=\ \int\frac{1}{\left(x\right)^2+\left(4\right)^2}\ dx$

$=\ \frac{1}{4}\tan^{-1}{\frac{x}{4}}+c$

Important

$\int\frac{1}{x^4+16}\ dx=\ \int\frac{1}{\left(x^2\right)^2+\ \left(4\right)^2}\ dx$

$=\ \frac{1}{4}\tan^{-1}{\ \frac{x^2}{4}}+c$

$\int\frac{1}{x^2+a^2}\ dx=\frac{1}{a}\tan^{-1}{\frac{x}{a}}+c\ \ \ \ \ \ here\ \left(x\right)^2\ is\ a\ variable$

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

$\int\frac{1}{\left(x^2\right)^2+\left(4\right)^2}\ dx^2\ =\ \frac{1}{4}\tan{\frac{x^2}{4}}+c$