Blog

Pattern – 4 : Indefinite Integration

No Comments

⋆ Pattern : 4

∫ $\left[f\left(x\right)\right]^n f' (x) dx = \frac{\left[f\left(x\right)\right]^{n+1\ }}{n+1}+c$

1) $\int\frac{x+2}{\sqrt{x^2+4x-7}}\ dx$

= $\ \frac{1}{2}\ \ \int{\left(x^2+4x-7\right)^\frac{-1}{2}\ \left(2x+4\right)}\ dx$

= $\ \frac{1}{2}\ \ \frac{\left(x^2+4x-7\right)^{\frac{1}{2}\ }}{\frac{1}{2}}+c$

= $\ \sqrt{x^2+4x-7}+c$

2) $\ \int\frac{\sqrt{1+\log{x}}}{x}\ dx$

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

$=\ \frac{\left(1+\log{x}\right)^\frac{3}{2}}{\frac{3}{2}}+c$

$=\ \frac{2\left(1+\log{x}\right)^\frac{3}{2}}{3}+c$