Integration By Parts- Via a TableTypically, integration by parts is introduced as:Zu dv = uv −Zv duWe want to be able to compute an integral using this method, but in a moreefficient way. Consider the following table:Zu dv ⇒+u dv− du vThe first column switches ± signs, the second column differentiates u, andthe third column antidifferentiates dv. We can write the result of integrationas multiplying the sign, +1 times u then going down along a diagonal andmultiplying by v. We then add the integral of the product going straightacross.Using this table, we can perform multiple integration by parts at onetime. Consider this example, with the corresponding table:Zt2e−3tdt ⇒+ t2e−3t− 2t (−1/3)e−3t+ 2 (1/9)e−3t− 0 (−1/27)e−3tUsing the same pattern as before, but continuing through, we see that evi-dently:Zt2e−3tdt = +t2(−1/3)e−3t+ (−2t)(1/9)e−3t+ 2(−1/27)e−3t++Z(−0 · (−1/27)e−3tdtSimplifying:Zt2e−3tdt = −e−3t13t2+29t +227Here are a couple more examples that usually require integration by parts:Zln(x) dx ⇒+ ln(x) 1− 1/x x1so that:Zln(x) dx = x ln(x) −Z1 dx = x ln(x) − xAnother example, where we integrate by parts twice to get a similarintegral on both sides of the equation:Ze−2tsin(3t) dt ⇒+ sin(3t) e−2t− 3 cos(3t) (−1/2)e−2t+ −9 sin(3t) (1/4)e−2tSo:Ze−2tsin(3t) dt = sin(3t)(−1/2)e−2t− 3 cos(3t)(1/4)e−2t+Z−9 sin(3t)(1/4)e−2tSimplifying:Ze−2tsin(3t) dt = −e−2t12sin(3t) +34cos(3t)−94Ze−2tsin(3t) dtNow solve for the integral:134Ze−2tsin(3t) dt = −e−2t12sin(3t) +34cos(3t)To finish, multiply both sides by 4/13.Extra Practice:Maple Commands are given so you can check your answer!1.Z√x ln(x) dx int(sqrt(x)*ln(x),x);2.Zx2cos(3x) dx int(x^2*cos(3*x),x);3.Zt3e−2tdt int(t^3*exp(-2*t),t);4.Ze−2tsin(2t) dt int(exp(-2*t)*sin(2*t),t);5.Ztan−1(1/t) dt