4. (8 points). Evaluate each integral. Show your work. a. ? t e? dt (integration by parts) b. ??² x ln x dx (integration by parts) c. ???/² sin³ x dx d. ? sin(2020x) cos(2x) dx
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J x ln x: We can use integration by parts with u = ln x and dv = x dx. Then, du = (1/x) dx and v = (1/2) x^2. Thus, J x ln x = (1/2) x^2 ln x - J (1/2) x dx = (1/2) x^2 ln x - (1/4) x^2 + C where C is the constant of integration. Show more…
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