1. Follow the steps below to find the indefinite integral of vx In(x). a) To compute / vx In(x) dx, we use integration by parts (the integral is the product of two functions vx and In(x)). Choose one of these functions to be u and the other to be -- using LIATE. du = dx du = b) Using the above choices, write the integral using the integration by parts formula (the only variable involved should be x): Vx In(x)dx = c) Solve the integral you found in (b) to find the indefinite integral of vx In(x)2. Evaluate the integral using integration by parts. Hints: LIATE, - (cos '(x)) = - 1 cos -1(x) dx 3. Evaluate the definite integral using integration by parts. Hint: Find an antiderivative and then evaluate it at the bounds of integration. -1/2 x cos(nx) dx4. For some integrals, you may need to use integration by parts more than once. Evaluate the integral (x2 + 3x + 1)e * dx5. Some integrals require a little extra cleverness to solve. Follow the steps to evaluate {use LlllTI-Z] I e\" sin(3x) mix a] Use integration by parts to rewrite the integral u = do = dx tiu = v = J's-z" sin[3x} tix = _I h] Simplify and rewrite the integral f e\" sin[3x} d1: = o] This integral is still not solvable using basic formulas. so we use integration lav parts again for the new integral in [h]. u = do = dx do. = 1;! = 3 3 __ 2x = _ _ _ 2 f a- maa) dx 2 J- ] d) Simplify and rewrite the integral 3 E J- an eosESx} = e] Combine [h] and [d] and simplify I an sin{3x} d1: = t] Notice in [d] we get back to the SAME integral we originally set out to solve! Let I = J. e1m sin(3x) nix and rewrite the equation in [e] using 1] [viewing it as a variable] and solve for 1]. This will he the integral we are looking for