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Pattern – 2 : Indefinite Integration

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⋆ Pattern – 2 :

If ∫f(x) dx = F(x) +c, then

∫ f (a + b) dx = $\frac{1}{a}$ F ( ax + b )dx + c : a ≠ 0

$\int\frac{1}{x}\ dx\ =\ log\ |\ x\ |\ +\ c$

$\int\ \frac{1}{5x-3}\ dx\ =\ \frac{1}{5}\ log\ |\ 5x\ - 3 \ | + c$

$\int\ \frac{1}{x^2+a^2}\ dx\ =\ \frac{1}{a}\ {tan}^{-1}{\frac{x}{a}}\ +\ c$

$\int\ sin\ x\ dx\ =\ -\ cos\ x\ +\ c$

$\int\frac{1}{5x^2+9}dx$

$=\ \int\frac{1}{\left(\sqrt5x\right)^2+\ \left(3\right)^2}\ dx$

$=\ \frac{1}{3}\times\frac{1}{\sqrt5}\ \tan^{-1}{\frac{\sqrt5x}{3}}+c$

$=\ \frac{1}{3\sqrt5}\tan^{-1}{\frac{\sqrt5x}{3}}+c$