Question

What is `int lnx/x^4 dx`?

Answer 1

`intfrac{ln(x)}{x^4}dx=-frac{3ln(x)+1}{9x^3}+c`,
where `c` is the constant of integration.

Explanation:

`intfrac{ln(x)}{x^4}dx=-1/3intln(x)frac{d}{dx}(x^{-3})dx`

`=-1/3[ln(x)\x^{-3}-intfrac{d}{dx}(ln(x))x^{-3}dx]`

`=-1/3ln(x)\x^{-3}+1/3int(1/x)x^{-3}dx]`

`=-frac{ln(x)}{3x^3}+1/3intx^{-4}dx`

`=-frac{ln(x)}{3x^3}+1/3frac{x^{-3}}{-3}+c`,
where `c` is the constant of integration.

`=-frac{3ln(x)+1}{9x^3}+c`