Question

∫ (logx)⁴ dx=.......+c,Select Proper option from the given options.
(a) (logx)⁵/5
(b) (logx)²/2
(c) logx⁵/5x
(d) logx.(logx)⁴+(logx)⁵/5x

Answer 1

use integration by part ,
ILATE is the key words to identify which 1st function and which is 2nd function.
here, f(x) = (logx)⁴ is 1st function
and g(x) = 1 is 2nd function.

now, use concepts $\int{fg}\,dx=f\int{g}\,dx-\int{\{f'\int{g}\,dx\}}\,dx$

$\int{(logx)^4}\,dx\\\\\\=(logx)^4\int{dx}-\int{4(logx)^3\frac{1}{x}x}\,dx\\\\\\=x(logx)^4-4\int{(logx)^3}\,dx\\\\\\=x(logx)^4-4[x(logx)^3-\int{3(logx)^2\frac{1}{x}x}\,dx]\\\\\\=x(logx)^4-4x(logx)^3+12\int{(logx)^2}\,dx\\\\\\=x(logx)^4-4x(logx)^3+12x(logx)^2-24\int{logx}\,dx\\\\\\=x(logx)^4-4x(logx)^3+12x(logx)^2-24x(logx)+24x\\\\\\=xlogx[(logx)^3-4(logx)^2+12(logx)-24]+24x$