Evaluate the indefinite integral integrate of (x^3(ln(x))dx)
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You need to integrate by parts, hence, you need to use the following formula such that:
`int f(x)g'(x)dx = f(x)g(x) - int f'(x)*g(x)dx`
Considering `f(x) = ln x` and `g'(x) = x^3` yields:
`f(x) = ln x => f'(x) = 1/x`
`g'(x) = x^3 => g(x) = int x^3 dx = x^4/4`
`int x^3*ln x dx = (x^4*ln x)/4 - int (1/x)(x^4/4)dx`
`int x^3*ln x dx = (x^4*ln x)/4 - (1/4)int x^3 dx ` `int x^3*ln x dx = (x^4*ln x)/4 - x^4/16 + c`
Factoring out `x^4/4` yields:
`int x^3*ln x dx = (x^4/4)(ln x -1/4) + c`
Hence, evaluating the given indefinite integral using parts yields `int x^3*ln x dx = (x^4/4)(ln x - 1/4) + c` .
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`int x^3 ln(x) dx`
To evaluate, use integration by parts. The formula is` int udv = uv - int vdu` .
So let,
`u = ln (x) ` and `dv= x^3 dx`
`du = 1/x dx ` `v=int x^3 dx=x^4/4`
Substitute u, v and du to the formula.
`int x^3 ln(x) dx = ln(x) * x^4/4 - int x^4/4 *1/xdx`
`int x^3 ln(x) dx=(x^4 ln(x))/4 - 1/4 int x^3 dx`
`int x^3 ln(x) dx=(x^4 ln(x))/4 - 1/4* x^4/4 + C`
`int x^3 ln(x) dx=(x^4 ln(x))/4 - x^4/16 + C`
Hence, `int x^3 ln(x) dx=(x^4 ln(x))/4 - x^4/16 + C` .