How do you find the integral of x^3 cos(x^2) dx? Calculus Techniques of Integration Integration by Parts 1 Answer Konstantinos Michailidis Sep 15, 2015 intx^3cos(x^2)dx=1/2x^2sin(x^2)+1/2cos(x^2) Explanation: We set t=x^2 hence dt=2xdx hence we have that int x^3cosx^2dx=int 1/2*x^2 cos(x^2) 2xdx=int1/2*tcostdt=1/2inttcostdt=1/2intt(sint)'dt=1/2[tsint]-1/2intt'sintdt=1/2tsint-1/2intsintdt=1/2tsint+1/2cost Hence intx^3cos(x^2)dx=1/2x^2sin(x^2)+1/2cos(x^2) Answer link Related questions How do I find the integral int(x*ln(x))dx ? How do I find the integral int(cos(x)/e^x)dx ? How do I find the integral int(x*cos(5x))dx ? How do I find the integral int(x*e^-x)dx ? How do I find the integral int(x^2*sin(pix))dx ? How do I find the integral intln(2x+1)dx ? How do I find the integral intsin^-1(x)dx ? How do I find the integral intarctan(4x)dx ? How do I find the integral intx^5*ln(x)dx ? How do I find the integral intx*2^xdx ? See all questions in Integration by Parts Impact of this question 2263 views around the world You can reuse this answer Creative Commons License