x sinx2dx x2 ex dxSolution d intergrate by substitution Vt

x sin(x2)dx x2 e-x dx

Solution

d) intergrate by substitution: V(t) = x sin(x^2) U=x^2 du=2x -> therefore (du/2)=x now we have; V(t) = [(1/2)sin(U)]du so the integral is: = -1/2 cos(U)+constant = -1/2 cos(x^2)+constant substitute the values.... e)int(X^2*e^(-x))dx Substitute s=-x ds=-1dx = - int(e^s*u^2) ds Integrate by parts u=s^2 dv=e^s ds du=2s ds v=e^s = int 2e^s * s ds - e^s * s^2 Factor out the 2: 2 int [ e^s * s ds - e^s s^2 Integrate by parts u=s dv=e^s ds du=1 ds v=e^s becomes = -e^s * s^2 * s - 2 int s^2 ds The integral of e^s is e^s becomes: = - e^s * s^2 + 2 e^s * s - 2 e^s + C Substitute back in s=-x = - e^(-s) * x^2 - 2 e^(-s) * x - 2 e^(-x) + C becomes = -e^(-x)[x^2 + 2x + 2] + C