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alt='integral x^3 log^2(x) dx = 1/32 x^4 (8 log^2(x) - 4 log(x) + 1) + constant'
title='integral x^3 log^2(x) dx = 1/32 x^4 (8 log^2(x) - 4 log(x) + 1) + constant'
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alt='Take the integral:
integral x^3 log^2(x) dx
For the integrand x^3 log^2(x), integrate by parts, integral f dg = f g - integral g df, where
f = log^2(x), dg = x^3 dx, df = (2 log(x))/x dx, g = x^4/4:
= 1/4 x^4 log^2(x) - 1/4 integral2 x^3 log(x) dx
Factor out constants:
= 1/4 x^4 log^2(x) - 1/2 integral x^3 log(x) dx
For the integrand x^3 log(x), integrate by parts, integral f dg = f g - integral g df, where
f = log(x), dg = x^3 dx, df = 1/x dx, g = x^4/4:
= -1/8 x^4 log(x) + 1/4 x^4 log^2(x) + 1/8 integral x^3 dx
The integral of x^3 is x^4/4:
= x^4/32 + 1/4 x^4 log^2(x) - 1/8 x^4 log(x) + constant
Which is equal to:
Answer: |
| = 1/32 x^4 (8 log^2(x) - 4 log(x) + 1) + constant'
title='Take the integral:
integral x^3 log^2(x) dx
For the integrand x^3 log^2(x), integrate by parts, integral f dg = f g - integral g df, where
f = log^2(x), dg = x^3 dx, df = (2 log(x))/x dx, g = x^4/4:
= 1/4 x^4 log^2(x) - 1/4 integral2 x^3 log(x) dx
Factor out constants:
= 1/4 x^4 log^2(x) - 1/2 integral x^3 log(x) dx
For the integrand x^3 log(x), integrate by parts, integral f dg = f g - integral g df, where
f = log(x), dg = x^3 dx, df = 1/x dx, g = x^4/4:
= -1/8 x^4 log(x) + 1/4 x^4 log^2(x) + 1/8 integral x^3 dx
The integral of x^3 is x^4/4:
= x^4/32 + 1/4 x^4 log^2(x) - 1/8 x^4 log(x) + constant
Which is equal to:
Answer: |
| = 1/32 x^4 (8 log^2(x) - 4 log(x) + 1) + constant'
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<info text='log(x) is the natural logarithm'>
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alt='log(x) is the natural logarithm'
title='log(x) is the natural logarithm'
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<link url='http://reference.wolfram.com/language/ref/Log.html'
text='Documentation'
title='Mathematica' />
<link url='http://functions.wolfram.com/ElementaryFunctions/Log'
text='Properties'
title='Wolfram Functions Site' />
<link url='http://mathworld.wolfram.com/NaturalLogarithm.html'
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alt='x^4 ((log^2(x))/4 - log(x)/8 + 1/32) + constant'
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alt='x^4/32 + 1/4 x^4 log^2(x) - 1/8 x^4 log(x) + constant'
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alt='Expand the following:
(x^4 (1 - 4 log(x) + 8 log(x)^2))/32
x^4 (1 - 4 log(x) + 8 log(x)^2) = x^4×1 + x^4 (-4 log(x)) + x^4×8 log(x)^2:
(x^4 - 4 x^4 log(x) + 8 x^4 log(x)^2)/32
(x^4 - 4 x^4 log(x) + 8 x^4 log(x)^2)/32 = x^4/32 - (4 x^4 log(x))/32 + (8 x^4 log(x)^2)/32:
x^4/32 - (4 x^4 log(x))/32 + (8 x^4 log(x)^2)/32
The gcd of -4 and 32 is 4, so (-4 x^4 log(x))/32 = ((4 (-1)) x^4 log(x))/(4×8) = 4/4×(-x^4 log(x))/8 = (-x^4 log(x))/8:
x^4/32 + (-1 x^4 log(x))/8 + (8 x^4 log(x)^2)/32
8/32 = 8/(8×4) = 1/4:
Answer: |
| x^4/32 - (x^4 log(x))/8 + (x^4 log(x)^2)/4'
title='Expand the following:
(x^4 (1 - 4 log(x) + 8 log(x)^2))/32
x^4 (1 - 4 log(x) + 8 log(x)^2) = x^4×1 + x^4 (-4 log(x)) + x^4×8 log(x)^2:
(x^4 - 4 x^4 log(x) + 8 x^4 log(x)^2)/32
(x^4 - 4 x^4 log(x) + 8 x^4 log(x)^2)/32 = x^4/32 - (4 x^4 log(x))/32 + (8 x^4 log(x)^2)/32:
x^4/32 - (4 x^4 log(x))/32 + (8 x^4 log(x)^2)/32
The gcd of -4 and 32 is 4, so (-4 x^4 log(x))/32 = ((4 (-1)) x^4 log(x))/(4×8) = 4/4×(-x^4 log(x))/8 = (-x^4 log(x))/8:
x^4/32 + (-1 x^4 log(x))/8 + (8 x^4 log(x)^2)/32
8/32 = 8/(8×4) = 1/4:
Answer: |
| x^4/32 - (x^4 log(x))/8 + (x^4 log(x)^2)/4'
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alt='integral_0^1 x^3 log^2(x) dx = 1/32 = 0.03125'
title='integral_0^1 x^3 log^2(x) dx = 1/32 = 0.03125'
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