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Pattern – 3 : Indefinite Integration

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⋆ Pattern – 3 : – Adjustment

1) $\int\frac{2x^3+18x-1}{x^2+9}\ dx$

$=\ \int\left[\frac{2x\left(x^2+9\right)}{\left(x^2+9\right)}-\ \frac{1}{x^2+9}\right]\ dx$

$=\ \int\left(2x-\ \frac{1}{x^2+9}\right)\ dx$

$=\ \frac{2x^2}{2}-\frac{1}{3}\tan^{-1}{\frac{x}{3}}+c$

$\int{x^n\ dx}=\ \frac{x^{n+1\ }}{n+1\ }+c$

2) $\int{x\ \sqrt{x+3}}\ dx$

= ∫ [ ( x + 3 ) $\sqrt{x+3} - 3 \sqrt{x+3} ] dx$

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

= $\ \frac{\left(x+3\right)^\frac{5}{2}}{\frac{5}{2}}-3\ \frac{\left(x+3\right)^\frac{3}{2}\ }{\frac{3}{2}}+c$

= $\ \frac{2\left(x+3\right)^\frac{5}{2}\ }{5}-2\left(x+3\right)^\frac{3}{2}+c$