# First make a substitution and then use integration by parts to evaluate the integral. integrate of x^3cos(x^2)dx

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Expert Answers

jeew-m | Certified Educator

Let;

`x^2 = t`

`2xdx = dt`

`xdx = (1/2)dt`

`int x^3cos(x^2) dx`

`= int x^2cos(x^2)*xdx`

`= int tcostdt/2`

`= 1/2int tcostdt`

Let;

`u = t`

`du = 1dt`

`v = sint`

`dv = costdt`

from integral by parts;

`int udv = uv-intvdu`

`int tcostdt = tsint-intsintdt`

`int tcostdt = tsint-(-cost)+C` where C is a constant.

`int x^3cos(x^2) dx = 1/2(x^2sin(x^2)+cos(x^2)+C)`