Integration by Parts Explained

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Questions and Answers

When applying integration by parts to $\int x^2 cos(x) dx$, what is the correct choice for u and $\frac{dv}{dx}$ based on LIATE?

$u = x^2, \frac{dv}{dx} = cos(x)$ (correct)
$u = cos(x), \frac{dv}{dx} = x^2$
$u = sin(x), \frac{dv}{dx} = x^2$
$u = x, \frac{dv}{dx} = x cos(x)$

After applying integration by parts once to $\int x^2 cos(x) dx$, which integral remains to be solved?

$\int sin(x) dx$
$\int 2x cos(x) dx$
$\int 2x sin(x) dx$ (correct)
$\int x^2 sin(x) dx$

To solve $\int x \sin x , dx$ using integration by parts, which of the following substitutions is most appropriate?

$u = x, dv = \sin x \, dx$ (correct)
$u = 1, dv = x \sin x \, dx$
$u = \sin x, dv = x \, dx$
$u = x \sin x, dv = dx$

What is the correct first step in evaluating the integral $\int x^3 \ln x , dx$ using integration by parts?

Let $u = \ln x$ and $dv = x^3 , dx$ (B) Signup and view all the answers

If $\int \ln(x) , dx = x\ln(x) - x + C$, what is the correct expression for $\int \ln^2(x) , dx$ after applying integration by parts once?

$x \ln^2(x) - 2 \int \ln(x) , dx$ (A) Signup and view all the answers

Which of the following integrals is correctly evaluated, assuming the constant of integration is omitted?

$\int tan(x) dx = ln |sec(x)|$ (B) Signup and view all the answers

Using the integration by parts formula, $\int u dv = uv - \int v du$, how should you select 'u' in the integral $\int x sin(x) dx$ according to the LIATE rule?

u = x (C) Signup and view all the answers

The formula for integration by parts is derived from which rule of differentiation?

The product rule (B) Signup and view all the answers

When applying integration by parts, the goal is to select 'u' and 'dv' such that:

$\int v du$ is easier to evaluate, while still being correct (B) Signup and view all the answers

According to the LIATE rule, which function should be prioritized as 'u' when integrating $\int x ln(x) dx$?

ln(x) because Logarithmic functions come first (A) Signup and view all the answers

Which of the following statements best describes a limitation of the integration by parts technique?

It is ineffective when both functions in the integrand become more complicated after differentiation or integration. (D) Signup and view all the answers

What is the correct application of the integration by parts formula to the integral $\int x e^x dx$?

$xe^x - \int e^x dx$ (A) Signup and view all the answers

What would be the next step to solve $\int x e^x dx$ after applying integration by parts and getting $xe^x - \int e^x dx$ ?

Solve $\int e^x dx$ directly. (C) Signup and view all the answers

When applying integration by parts, how does the LIATE rule guide the selection of $u$ and $dv$?

It suggests choosing $u$ based on the order: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential, prioritizing functions that simplify when differentiated. (B) Signup and view all the answers

For the integral $\int x \cos(x) dx$, what are the appropriate choices for $u$ and $dv$ when using integration by parts?

$u = x$, $dv = \cos(x) dx$ (A) Signup and view all the answers

What is the correct setup for integration by parts applied to $\int x^3 \ln(x) dx$?

$u = \ln(x), dv = x^3 dx$ (A) Signup and view all the answers

Given $\int x^2 e^{2x} dx$, what is the result after the first application of integration by parts?

$\frac{1}{2}x^2e^{2x} - \int xe^{2x} dx$ (A) Signup and view all the answers

In evaluating $\int x^2 e^{2x} dx$, after applying integration by parts twice, what is the correct form of the final integral (before adding the constant of integration)?

$\frac{1}{2}x^2e^{2x}-\frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x}$ (D) Signup and view all the answers

To solve $\int x^2 \cos(x) dx$, how many times would you typically need to apply integration by parts?

Twice (B) Signup and view all the answers

What is the primary reason for rewriting $\int x^3 \ln(x) dx$ as $\int \ln(x) \cdot x^3 dx$ before applying integration by parts?

To align with the LIATE rule, placing the logarithmic function in the 'u' position. (B) Signup and view all the answers

If you incorrectly choose $u$ and $dv$ when applying integration by parts, what is the most likely outcome?

The resulting integral will be more complex than the original. (B) Signup and view all the answers

Flashcards

Integration by Parts

A technique to integrate the product of two functions. It reverses the product rule of differentiation.

LIATE

A guideline for choosing 'u' in integration by parts: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential.