For ∫(2x+1)e^(3x) dx, take u = 2x+1 and dv = e^(3x) dx. Which choice of v has derivative e^(3x)?

• v = e^(3x)/3
• v = 3e^(3x)
• v = e^(3x)
• v = e^(3x)/9

Which equation is the integration by parts formula?

• Integral of u dv = u v + integral of v du
• Integral of u dv = u v - integral of v du
• Integral of u dv = integral of v du - u v

For x > 0, which split turns ∫x² ln x dx into a product minus a power integral containing no logarithm?

• u = x², dv = ln x dx
• u = 1, dv = x² ln x dx
• u = ln x, dv = x² dx

For ∫x e^x dx, which split lowers the polynomial degree and uses a known exponential antiderivative for v?

• u = x, dv = e^x dx
• u = e^x, dv = x dx
• u = x e^x, dv = dx

Using radians, let I = ∫e^(2x)sin x dx and J = ∫e^(2x)cos x dx. Parts gives I = e^(2x)sin x/2-J/2. Which second relation follows by taking u = cos x in J? Relations are up to constants.

• J = e^(2x)cos x/2-I/2
• J = e^(2x)sin x/2+I/2
• J = e^(2x)cos x/2+I/2
• J = e^(2x)cos x/2+2I

Using radians, let I = ∫e^x cos x dx. Two parts applications give I = e^x(cos x+sin x)-I, up to a constant. Which step isolates I?

• Subtract I from both sides, then divide by 2
• Add I to both sides, then divide by 2
• Add I to both sides, then divide by -2

Using radians, what is ∫x cos x dx?

• x sin x + cos x + C
• x sin x - cos x + C
• x cos x + sin x + C

Let f have a continuous derivative on [0,2], with f(0) = 3, f(2) = 4 and ∫ from 0 to 2 of f(x) dx = 5. Find ∫ from 0 to 2 of (2x+1)f'(x) dx.

• 13
• 7
• 17
• -7

Using the polynomial as u and dv = e^x dx each round, how many parts applications reduce ∫x²e^x dx to a plain exponential integral?

• 1
• 2
• 3

For x > 0, what is ∫(ln x)² dx after repeated integration by parts?

• x(ln x)² + 2x ln x + 2x + C
• x(ln x)² - 2x ln x - 2x + C
• x(ln x)² - 3x ln x + 2x + C
• x(ln x)² - 2x ln x + 2x + C