# Evaluate the indefinite integral integrate of (x^3(ln(x))dx)

sciencesolve | Certified Educator

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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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Jedidiah Hahn | Certified Educator

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starTop subjects are Math and Science

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  .

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