Trade one integral for an easier one

Mathematics · Calculus & Analysis · ages 19-20

Name ______________________ Date ____________

To integrate x times e to the x, which split into u and dv is the right start?
- u = x, dv = e to the x dx
- u = e to the x, dv = x dx
- u = x e to the x, dv = dx
- u = dx, dv = x e to the x
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 = v du - u
To integrate x^2 times ln x, which choice of u is best?
- u = x^2
- u = dx
- u = ln x
To integrate x times e^x, which split is the right start?
- u = x, dv = e^x dx
- u = e^x, dv = x dx
- u = x e^x, dv = dx
While integrating e to the x times cos x, the original integral reappears on the right side. What now?
- Collect it with the left side and solve for it
- Start over with u and dv swapped
- Conclude the integral does not exist
- Replace cos x with sin x and repeat
While integrating e^x times cos x, the original integral reappears on the right. What now?
- Stop, since it cannot be finished
- Collect it on the left and solve for it
- Switch to a substitution instead
What is the integral of x cos x dx?
- x sin x + cos x + C
- x sin x - cos x + C
- x cos x + sin x + C
What is the integral of x cos x dx?
- x sin x + cos x + C
- x sin x minus cos x + C
- x cos x + sin x + C
- x cos x minus sin x + C
How many full applications does x^2 times e^x need before the polynomial is gone?
- 1
- 2
- 3
To integrate x squared times ln x, which choice of u is best?
- u = ln x, dv = x squared dx
- u = x squared, dv = ln x dx
- u = x squared ln x, dv = dx
- u = dx, dv = x squared ln x

Answer key

For grown-ups. Fold this page away before handing over the rest.

Trade one integral for an easier one W1-mt_v6jZVs2_tf-s1

u = x, dv = e to the x dx · Differentiating x gives 1, which simplifies the product, while e to the x is easy to integrate.
Integral of u dv = u v - integral of v du · The product rule read backwards gives u v minus the integral of v du.
u = ln x · LIATE ranks logs first, so ln x becomes u and the rest becomes dv.
u = x, dv = e^x dx · LIATE puts algebra before exponentials, so x becomes u and e^x dx becomes dv.
Collect it with the left side and solve for it · Two rounds return a multiple of the original integral, so treat it as an unknown and solve the linear equation.
Collect it on the left and solve for it · Treat the returning integral as an unknown and solve for it with algebra.
x sin x + cos x + C · With u = x and dv = cos x dx, you get x sin x plus cos x, plus C.
x sin x + cos x + C · With u equals x and dv equals cos x dx, you get x sin x minus the integral of sin x, which is x sin x plus cos x.
2 · Each round drops the power by one, so x^2 needs two rounds.
u = ln x, dv = x squared dx · Logarithms simplify beautifully under differentiation, and x squared is trivial to integrate.