∫ (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
k4kunal09pe7o1t:
integration e^t .t^5 dt. = and put t =log x ......option d is correct
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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{(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](https://tex.z-dn.net/?f=%5Cint%7B%28logx%29%5E4%7D%5C%2Cdx%5C%5C%5C%5C%5C%5C%3D%28logx%29%5E4%5Cint%7Bdx%7D-%5Cint%7B4%28logx%29%5E3%5Cfrac%7B1%7D%7Bx%7Dx%7D%5C%2Cdx%5C%5C%5C%5C%5C%5C%3Dx%28logx%29%5E4-4%5Cint%7B%28logx%29%5E3%7D%5C%2Cdx%5C%5C%5C%5C%5C%5C%3Dx%28logx%29%5E4-4%5Bx%28logx%29%5E3-%5Cint%7B3%28logx%29%5E2%5Cfrac%7B1%7D%7Bx%7Dx%7D%5C%2Cdx%5D%5C%5C%5C%5C%5C%5C%3Dx%28logx%29%5E4-4x%28logx%29%5E3%2B12%5Cint%7B%28logx%29%5E2%7D%5C%2Cdx%5C%5C%5C%5C%5C%5C%3Dx%28logx%29%5E4-4x%28logx%29%5E3%2B12x%28logx%29%5E2-24%5Cint%7Blogx%7D%5C%2Cdx%5C%5C%5C%5C%5C%5C%3Dx%28logx%29%5E4-4x%28logx%29%5E3%2B12x%28logx%29%5E2-24x%28logx%29%2B24x%5C%5C%5C%5C%5C%5C%3Dxlogx%5B%28logx%29%5E3-4%28logx%29%5E2%2B12%28logx%29-24%5D%2B24x "\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")
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
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