Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.a. The integral ? x2e2x dx can be evaluated analytically using integration by parts.________________b. To evaluate the integral analytically, it is best to use partial fractions.________________c. One computer algebra system produces ? 2 sin x cos x dx =sin2 x. Another computer algebra system produces ? 2sin x cos x dx = ?cos2 x. One computer algebra system is wrong (apart from a missing constant of integration).
Solution 1RE We need to explain whether the given statements are true or false. STEP 1 (a). Integration by parts: udv = uv vd, provided both u and v are differentiable functions. Here the given integral is x e dx 2 2x Therefore let u = x and dv = e dx Thus we get du = 2xdx and v = e2x 2 Therefore 2 2x 2e2x e2x 2e2x 2x x e dx = x 2 2 2xdx = x 2 e dx By using integration by parts for xe x we get Therefore let u = x and dv = e dx 2x Thus we get du = 1 and v = e2x 2 Therefore 2x 2x xe dx = x e2 e2 x 2e2x(x e2xe 2xx2) = x (x(1x) = x (x1+x) 2 2 2 2 2 ex x 22x1) 2 2x Thus the given statement “The integral x e dx can be evaluated analytically using integration by parts.” is true. STEP 2 (b). The given statement is “to evaluate the integral 21 dxanalytically,it is best to use partial x 100 fractions”. 1 cannot be written in terms of partial fractions. x 100 Therefore the given statement is false. STEP 3 (c). The given statemen t is”. One computer algebra system produces 2 sin x cos x dx =sin2 x . Another computer algebra system produces 2sin cos x dx = cos2 x . One computer algebra system is wrong (apart from a missing constant of integration).” 2 sin(x)cos(x) = sin2x) =cos x+C 2 Therefore one computer algebra system is wrong.