for 5-12 it asks to \"identify u and dv for finding the integral uing integratio

Question

for 5-12 it asks to "identify u and dv for finding the integral uing integration by part"
5. ? xe^2x dx
6. ? x^2 e^2x dx
7. ? (ln x)^2 dx
8. ? ln3x dx 9. ? x sec^2 x dx
10. ? x^2 cos x dx
11. ? xe^-2x dx
12. ? 2x/e^x dx

for 27 and 28 " find the integral"
27. ? x cos x dx
28. ? x sinx dx

Explanation / Answer

27)int udv = uv- int vdu intgral of cox = sin x cosx = d(sinx)/dx int xcosx dx = int x d(sinx) = x.sinx - int(sinx.1. dx) = x sinx - int(sinxdx) = xsinx + cos x(integral of sin x = -cos x) +c where c is arbritary constant 28)? x sinx dx This uses integration by parts: uv - ? v du u = x du = dx dv = sinx v = -cosx Now plug in: -xcosx + ? cosx dx -xcosx + sinx + C 11)? xe^-2x dx = x ?e^(-2x) dx - ?[d/dx(x) ?e^(-2x)dx]dx = - (x/2)e^(-2x) + (1/2) ?e^(-2x) dx = - (x/2)e^(-2x) - (1/4) e^(-2x) + c = - (1/4)(2x + 1)e^(-2x) + c. 10) ? x^2 cos (x) dx integrate by parts twice let u = x^2 and dv = cos(x) dx du = 2x dx and v = sin (x) ? x^2 cos (x) dx = x^2 sin(x) - ?2x sin (x) dx let u = 2x and dv = sin (x) dx du = 2 dx and v = - cos (x) ? x^2 cos (x) dx = x^2 sin(x) - [ - 2x cos (x) - ?- 2 cos(x) dx ] = x^2 sin(x) + 2x cos (x) - 2?cos(x) dx = x^2 sin(x) + 2x cos (x) - 2 sin(x) + C 7) integrate((lnx)^2*dx) x = e^t >>>> dx = e^t integrate(t^2*e^t)*dt integrating part by part : t^2 = u >>>> du = 2t*dt e^t = dv >>>> v = e^t u*v - integrate(v*du) t^2*e^t - 2integrate(e^t*tdt) .....(*) Now, let's integrate : integrate(e^t*t*dt) u = t >>>>>> du = dt e^t = dv >>>> v = e^t e^t*t - integrate(e^t*dt) = e^t*t - e^t replacing of (*) t^2*e^t - 2integrate(e^t*tdt) t^2*e^t - 2e^t*t + 2e^t 2e^t + t^2*e^t - 2e^t*t remember : x = e^t 2x + (lnx)^2*x - 2x*lnx 6) integration by parts as you say. u = x^2 du/dx = 2x dv/dx = e^2x v = (1/2)e^2x looks like we gonna have to do it by parts twice. (x^2 / 2)e^2x - int.xe^2x now do by parts again. u = x du/dx = 1 dv/dx = e^2x v = (1/2)e^2x (x^2 / 2)e^2x - (x/2)e^2x + int.((1/2)e^2x) (x^2 / 2)e^2x - (x/2)e^2x + (1/4)e^2x + c 5) Use integration by parts. Let u = x du =dx v = e^2x Hence (xe^2x = xe^2x - ((e^2x)dx (xe^2x = xe^2x - ((e^2x)(2/2)dx (xe^2x = xe^2x - (1/2)e^2x + C where C = constant of integration

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