Which of the following is the result of applying integration by parts once to the integral $$\int e^x \sin(x) dx$$?
The parts formula, LIATE strategy, tabular method, and repeated integration by parts.
66 questions
Which of the following is the result of applying integration by parts once to the integral $$\int e^x \sin(x) dx$$?
$$e^x \sin(x) + \int e^x \cos(x) dx$$
$$e^x \cos(x) - \int e^x \sin(x) dx$$
$$e^x \sin(x) - \int e^x \cos(x) dx$$
$$-e^x \cos(x) + \int e^x \sin(x) dx$$
Which of the following describes a common error made when applying integration by parts?
Using the wrong limits of integration
Incorrect choice of 'u' and 'dv'
Integrating 'u' instead of differentiating
Forgetting to differentiate 'dv'
What is the value of $$\int_{0}^{1} x \sin(x) dx$$?
$$\sin(1)$$
$$1 + \cos(1)$$
$$1 - \cos(1)$$
$$\cos(1) - 1$$
Let $$g$$ be a function with a continuous derivative. Given that $$g(1) = 2$$, $$g(3) = 8$$, and $$\displaystyle\int_{1}^{3} g(x) dx = 12$$, what is the value of $$\displaystyle\int_{1}^{3} (x-1) \cdot g'(x) dx$$?
$$6$$
$$10$$
$$16$$
$$4$$
Let $$v$$ be a function with a continuous derivative. Given that $$v(1) = 5$$, $$v(4) = 11$$, and $$\displaystyle\int_{1}^{4} v(x) dx = 18$$, what is the value of $$\displaystyle\int_{1}^{4} (x-3) \cdot v'(x) dx$$?
$$21$$
$$6$$
$$39$$
$$3$$
Let $$r$$ be a function with a continuous derivative. If $$r(1) = 6$$, $$r(4) = 12$$, and $$\int_{1}^{4} r(x) dx = 21$$, what is the value of $$\int_{1}^{4} (x-2) r'(x) dx$$?
$$15$$
$$33$$
$$9$$
$$27$$
Let $$m$$ be a function with a continuous derivative. If $$m(-1) = 3$$, $$m(2) = 9$$, and $$\displaystyle\int_{-1}^{2} m(x) dx = 12$$, what is the value of $$\displaystyle\int_{-1}^{2} (2x+1) m'(x) dx$$?
$$24$$
$$36$$
$$30$$
$$18$$
Let $$f$$ be a function with a continuous derivative such that $$f(0) = 1$$, $$f(2) = 5$$, and $$\int_{0}^{2} f(x) dx = 7$$. What is the value of $$\int_{0}^{2} x \cdot f'(x) dx$$?
$$3$$
$$10$$
$$17$$
$$6$$
Use integration by parts to evaluate $$\int_{1}^{3} x \ln(x) dx$$.
$$\frac{9}{4} \ln(3) - 1$$
$$\frac{9}{4} \ln(3) - 2$$
$$\frac{9}{2} \ln(3) - 1$$
$$\frac{9}{2} \ln(3) - 2$$
What is the value of $$\int_{0}^{\frac{\pi}{2}} x e^{\sin(x)} \cos(x) dx$$?
$$e - \frac{\pi}{2}$$
$$\frac{\pi}{2} e - \int_{0}^{\frac{\pi}{2}} e^{\sin(x)} dx$$
$$e - 1$$
$$\frac{\pi}{2}$$
Use integration by parts to evaluate $$\int_{0}^{1} x^2 e^x dx$$.
$$2e + 3$$
$$e^2 - 2$$
$$2e - 3$$
$$3e - 2$$
What is the value of $\displaystyle\int_{0}$^1 e^x cos(x) dx?
$$\frac{e·sin(1) + e·cos(1) - 1}{2}$$
$$e·cos(1) - 1$$
$$e·sin(1)$$
$$\frac{e·sin(1) - e·cos(1) + 1}{2}$$
Evaluate $$\int_{1}^{2} x \ln(x) dx$$ using integration by parts.
$$\frac{1}{2} - \ln(2)$$
$$\frac{1}{2} + \ln(2)$$
$$\frac{3}{4} - \ln(2)$$
$$\frac{3}{4} + \ln(2)$$
What is the value of $$\int_{0}^{\pi/2} x \cos(x) dx$$?
$$0$$
$$\frac{\pi}{2}$$
$$1$$
$$\frac{\pi}{4}$$
Which of the following integrals must be evaluated using integration by parts?
$$\int x e^x dx$$
$$\int x^2 dx$$
$$\int e^x dx$$
$$\int \sin x dx$$
Which of the following integrals requires integration by parts for evaluation?
$$\int_{a}^{b} 30x^3+20x^2$$
$$\int_{a}^{b} 2x+5$$
$$\int_{a}^{b} 3x^2+3$$
$$\int_{a}^{b} e^x cos(x)$$
Which of the following integrals is best evaluated using integration by parts?
$$\int$$sin(x)cos(x) dx
$$\int$$e^(3x) dx
$$\int$$x ln(x) dx
$$\int$$(x^2 + 1)^5 dx
Which of the following is equal to $\displaystyle\int e^x \sin(x) dx$?
$$\displaystyle\frac{e^x}{2}(\sin(x) + \cos(x)) + C$$
$$\displaystyle e^x \sin(x) - e^x \cos(x) + C$$
$$\displaystyle\frac{e^x}{2}(\sin(x) - \cos(x)) + C$$
$$\displaystyle e^x \sin(x) + e^x \cos(x) + C$$
Use integration by parts to evaluate $$\int$$x e^x dx.
x * e^x - e^x + C
x - e^x + C
x * e^x - e + C
x * e^x - x + C
Evaluate $$\int$$x²e^x dx.
x²e^x - 2xe^x + e^x + C
x²e^x - 2xe^x - 2e^x + C
x²e^x - 2xe^x + 2e^x + C
x²e^x - xe^x + e^x + C
Use integration by parts to evaluate $$\int$$x * e^(2x) dx.
(1/2) * x * e^(2x) - e^(2x) + C
(1/2) * x * e^(2x) - (1/4) * e^(2x) + C
(1/2) * x * e^(2x) - (1/4) * e^(4x) + C
(1/2) * x * e^(2x) - (1/2) * e^(2x) + C
Which of the following is equivalent to $$\int x e^x dx$$?
$$xe^x - e^x + C$$
$$xe^x + e^x + C$$
$$e^x - xe^x + C$$
$$e^x + xe^x + C$$
$$\displaystyle\int 2x \sin(3x) dx =$$
$$\displaystyle\frac{2x}{3} \cos(3x) + \frac{2}{9} \sin(3x) + C$$
$$\displaystyle-\frac{2}{3} \cos(3x) + \frac{2x}{9} \sin(3x) + C$$
$$\displaystyle-\frac{2x}{3} \cos(3x) + \frac{2}{3} \sin(3x) + C$$
$$\displaystyle-\frac{2x}{3} \cos(3x) + \frac{2}{9} \sin(3x) + C$$
Evaluate the indefinite integral $$\displaystyle\int x^2 e^x dx$$.
$$\displaystyle e^x(x^2 + 2x + 2) + C$$
$$\displaystyle e^x(x^2 - 2x + 2) + C$$
$$\displaystyle e^x(x^2 - 2x - 2) + C$$
$$\displaystyle x^2 e^x - 2x e^x + C$$
$$\displaystyle\int x \sec^2(x) dx =$$
$$\displaystyle x \tan(x) + \ln|\cos(x)| + C$$
$$\displaystyle x \tan(x) - \sec(x) + C$$
$$\displaystyle x \tan(x) - \ln|\cos(x)| + C$$
$$\displaystyle\frac{x^2}{2} \sec^2(x) + x \tan(x) + C$$
What is the value of $\displaystyle\int_{0}^{\pi/2} x \sin(x) dx$?
$$\displaystyle 1$$
$$\displaystyle\frac{\pi}{2}$$
$$\displaystyle\frac{\pi}{2} - 1$$
$$\displaystyle\frac{\pi}{4}$$
Which of the following is $\displaystyle\int x^2 \cos(x) dx$?
$$\displaystyle (x^2 - 2) \sin(x) + 2x \cos(x) + C$$
$$\displaystyle x^2 \sin(x) + 2x \cos(x) + 2 \sin(x) + C$$
$$\displaystyle x^2 \sin(x) - 2x \cos(x) + C$$
$$\displaystyle \frac{x^3}{3} \sin(x) - x^2 \cos(x) + C$$
$$\int x \cos(2x) dx =$$
$$\dfrac{1}{2} x \sin(2x) - \dfrac{1}{4} \cos(2x) + C$$
$$\dfrac{1}{2} x \sin(2x) + \dfrac{1}{4} \cos(2x) + C$$
$$\dfrac{1}{2} x^2 \sin(2x) + C$$
$$\dfrac{1}{2} x^2 \cos(2x) + \dfrac{1}{2} \sin(2x) + C$$
Evaluate $$\int$$x sin(x) dx.
x * cos(x) - cos(x) + C
x * cos(x) + cos(x) + C
-x * cos(x) + sin(x) + C
x * cos(x) + sin(x) + C
$$\int$$ln(x)cos(x) dx =
ln(x)sin(x) - sin(x) + C
ln(x)sin(x) + $$\int$$sin(x)/x dx + C
ln(x)sin(x) - cos(x)/x + C
ln(x)sin(x) - $$\int$$sin(x)/x dx + C
An additional algebraic step is required to evaluate an integral using the method of integration by parts when the process results in which of the following?
When using integration by parts produces an algebraic function
When using integration by parts yields a trigonometric function
When using integration by parts initially produces the original integral again
When using integration by parts yields a polynomial
Which of the following best describes the relationship between the technique of integration by parts and the product rule for differentiation?
Integration by parts and the product rule are unrelated techniques used in different branches of calculus.
The product rule is derived from the integration by parts formula, showcasing their inverse relationship.
Integration by parts is the reverse process of the product rule, allowing us to integrate products of functions.
Integration by parts is an alternative to the product rule when differentiating complex functions.
Which of the following statements best describes the relationship between the formula for integration by parts and the product rule for differentiation?
The integration by parts formula is a special case of the product rule when one function is constant.
Integration by parts is the reverse process of the product rule, derived by integrating both sides of the product rule equation.
Integration by parts is an extension of the product rule, applying it to integrals of products of functions.
Integration by parts and the product rule are unrelated techniques used in different branches of calculus.
Evaluate $$\int_{1}^{e} x \ln(x) dx$$.
$$(e^2 + 1)/4$$
e²/2 - e/2 - 1/4
e·ln(e) - 1·ln(1) - (e-1)
(e²ln(e) - 1)/2 - (e-1)
When applying integration by parts, what is the primary guideline for selecting a factor $g(x)$ from the integrand?
For g(x), choose whichever portion is easiest to integrate
For g(x), selecting the portion of the integrand that contains a cubic function will work
For g(x), selecting any portion of the integrand can work
For g(x), choose whichever portion is easiest to take the derivative of
Which of the following is the formula for integration by parts?
$$\int u dv = uv + \int v du$$
$$\int u dv = uv - \int v du$$
$$\int u dv = uv$$
$$\int u dv = \int v du$$
When applying the method of integration by parts, which principle best guides the selection of the portion of the integrand designated as $f(x)$?
For f(x), selecting any portion of the integrand can work
For f(x), choose whichever portion is easiest to take the derivative of
For f(x), choose whichever portion is a polynomial
For f(x), choose whichever portion is easiest to take the integral of
Let $$f$$ be a differentiable function satisfying the equation $$\displaystyle \int f(x)\sin x dx = -f(x)\cos x + \int 4x^3\cos x dx$$. Which of the following could be $$f(x)$$?
$$x^4$$
$$4x^3$$
$$\cos x$$
$$\sin x$$
When using integration by parts for an integrand that includes a polynomial factor, what is the recommended assignment for that factor?
The portion of the integral that is a polynomial should be your g(x)
The portion of the integral that is a polynomial should be your f(x).
The portion of the integral that is a polynomial is not included in the integration by parts formula.
The portion of the integral that is a polynomial should be ignored when using integration by parts
If integration by parts is used to evaluate the integral $$\int x \cos x dx$$, which of the following is the correct choice for $$u$$ and $$dv$$?
$$u = \cos x$$, $$dv = x dx$$
$$u = \sin x$$, $$dv = x dx$$
$$u = x$$, $$dv = \cos x dx$$
$$u = x^2$$, $$dv = \sin x dx$$
When applying the method of integration by parts, which of the following function types is generally the most strategic choice for 'u'?
A constant function
A logarithmic function
A trigonometric function
An exponential function
According to general guidelines for integration by parts, should the trigonometric factor of an integrand be designated as `f(x)` (the part to be differentiated) or `g(x)` (the part to be integrated)?
The portion of the integral that is a trigonometric function should be your f(x).
The portion of the integrand that contains a trigonometric function should be ignored.
The portion of the integral that is a trigonometric function is not included in the integration by parts formula.
The portion of the integrand that contains a trigonometric function should be your g(x).
The table above gives selected values for the twice-differentiable functions $$f$$ and $$g$$, and for the first and second derivatives of $$g$$:
| $$x$$ | 2 | 4 |
|---|---|---|
| $$f(x)$$ | 7 | 13 |
| $$g(x)$$ | 2 | 9 |
| $$g'(x)$$ | 1 | 7 |
| $$g''(x)$$ | 5 | 8 |
Given that $$f'(x) = 3$$ for all $$x$$, what is the value of $$\displaystyle \int_{2}^{4} f(x)g''(x) dx$$?
69
63
84
78
Which of the following is equivalent to $$\int x\ln(x) dx$$?
$$\frac{x^2}{2}*\ln(x) - \frac{x^2}{4} + C$$
$$\frac{x^2}{2}*\ln(x) + \frac{x^2}{4} + C$$
$$x*\ln(x)-x+ C$$
$$\ln\left(\frac{x^2}{2}\right) - \frac{x^2}{4} + C$$
| $$x$$ | $$f(x)$$ | $$f'(x)$$ |
|---|---|---|
| $$0$$ | $$2$$ | $$5$$ |
| $$4$$ | $$-3$$ | $$11$$ |
The function $$f$$ has a continuous derivative. The table above provides values for $$f(x)$$ and its derivative, $$f'(x)$$, at $$x=0$$ and $$x=4$$. If $$\displaystyle \int_{0}^{4} f(x) dx = 8$$, what is the value of $$\displaystyle \int_{0}^{4} x f'(x) dx$$?
$$-13$$
$$-20$$
$$-12$$
$$-7$$
Which of the following is equivalent to the indefinite integral $$\displaystyle\int x \arctan(x) dx$$?
$$\displaystyle\frac{x^2 + 1}{2} \arctan(x) - \frac{x}{2} + C$$
$$\displaystyle\frac{x^2}{2} \arctan(x) - \frac{x}{2} + C$$
$$\displaystyle\frac{x^2}{2} \arctan(x) + \frac{1}{2} \ln(1 + x^2) + C$$
$$\displaystyle\frac{x^2 - 1}{2} \arctan(x) + \frac{x}{2} + C$$
Let the functions `f`, `g`, and `h` be defined for `0 ≤ x ≤ 4` by `f(x) = x²`, `g(x) = $\displaystyle\int_{0}$^x f(t) dt`, and `h(x) = g(4) - g(x)`. Which of the following is equivalent to `$\displaystyle\int_{0}$^4 h(x) dx`?
$\displaystyle\int_{0}$^4 xf(x)dx
g(4)
4g(4) - $\displaystyle\int_{0}$^4 g(x)dx
$\displaystyle\int_{0}$^4 f(x)dx
Let $$t$$ be a function with a continuous derivative. Given that $$t(0) = 4$$, $$t(3) = 10$$, and $$\displaystyle\int_{0}^{3} t(x) dx = 15$$, what is the value of $$\displaystyle\int_{0}^{3} (5-x) \cdot t'(x) dx$$?
$$105$$
$$15$$
$$6$$
$$-15$$
Let $$k$$ be a function with a continuous derivative. If $$k(0) = 1$$, $$k(3) = 7$$, and $$\displaystyle\int_{0}^{3} k(x) dx = 9$$, what is the value of $$\displaystyle\int_{0}^{3} (3x-2) \cdot k'(x) dx$$?
$$24$$
$$33$$
$$18$$
$$42$$
Let $$h$$ be a function with a continuous derivative. Given that $$h(0) = 3$$, $$h(2) = 7$$, and $$\displaystyle\int_{0}^{2} h(x) dx = 8$$, what is the value of $$\displaystyle\int_{0}^{2} (x+1) h'(x) dx$$?
$$10$$
$$18$$
$$4$$
$$29$$
Let $$w$$ be a function with a continuous derivative. Given that $$w(-2) = 5$$, $$w(1) = 11$$, and $$\displaystyle\int_{-2}^{1} w(x) dx = 18$$, what is the value of $$\displaystyle\int_{-2}^{1} x^2 w'(x) dx$$?
$$-30$$
$$-18$$
$$42$$
$$6$$
Let $$s$$ be a function with a continuous derivative such that $$s(0) = 2$$, $$s(4) = 8$$, and $$\displaystyle\int_{0}^{4} s(x) dx = 20$$. What is the value of $$\displaystyle\int_{0}^{4} (4x-3) \cdot s'(x) dx$$?
$$30$$
$$24$$
$$36$$
$$48$$
Let $$p$$ be a function with a continuous derivative. Given that $$p(-1) = 4$$, $$p(2) = 10$$, and $$\int_{-1}^{2} p(x) dx = 15$$, what is the value of $$\int_{-1}^{2} (2x-1)p'(x) dx$$?
$$18$$
$$30$$
$$24$$
$$12$$
$$\displaystyle\int x \ln(x) dx =$$
$$\displaystyle\frac{x^2}{4}(2 \ln(x) - 1) + C$$
$$\displaystyle x^2 \ln(x) - \frac{x^2}{4} + C$$
$$\displaystyle\frac{x^2}{2} \ln(x) + \frac{x}{2} + C$$
$$\displaystyle\frac{x^2}{2} \ln(x) - \frac{x^2}{2} + C$$
The table below gives selected values for a twice-differentiable function $$f$$ and its derivatives.
| $$x$$ | $$f(x)$$ | $$f'(x)$$ | $$f''(x)$$ |
|---|---|---|---|
| $$0$$ | $$1$$ | $$-2$$ | $$5$$ |
| $$1$$ | $$2$$ | $$6$$ | $$-1$$ |
What is the value of $$\displaystyle\int_{0}^{1} x f''(x) dx$$?
6
5
-1/2
-1
What is $$\int$$xe^(-2x)sin(3x) dx?
$$-xe^(-2x)cos(3x)/3 + xe^(-2x)sin(3x)/2 - e^(-2x)cos(3x)/6 - e^(-2x)sin(3x)/4 + C$$
$$-xe^(-2x)cos(3x)/3 - xe^(-2x)sin(3x)/2 + e^(-2x)cos(3x)/6 - e^(-2x)sin(3x)/4 + C$$
$$xe^(-2x)cos(3x)/3 - xe^(-2x)sin(3x)/2 + e^(-2x)cos(3x)/6 + e^(-2x)sin(3x)/4 + C$$
$$-\frac{x e^{-2x}(2\sin(3x)+3\cos(3x))}{13}+\frac{e^{-2x}(5\sin(3x)-12\cos(3x))}{169}+C$$
In which of the following integrals does applying the LIATE mnemonic to select 𝑢 for integration by parts lead to the most straightforward evaluation?
$$\int \tan(x) dx$$
$$\int e^x \cos(x) dx$$
$$\int x^2 \sin(x) dx$$
$$\int x \ln(x) dx$$
How many times should the integration by parts procedure be applied to evaluate an integral?
As many times as needed in order to keep simplifying the resulting integrand
Two times only
One time only
At least three times
Let h(x) = $$\int$$e^x sin(x) dx. If h(0) = 3, then h(x) =
e^x(sin(x) - cos(x))/2 + 3
e^x·sin(x) + 3
e^x(sin(x) - cos(x))/2 + C
e^x(sin(x) - cos(x))/2 + 3 + 1/2
Which of the following best describes the primary purpose of using integration by parts?
To simplify complex integrals
To find limits of functions
To evaluate definite integrals
To solve differential equations
Which of the following is equivalent to the indefinite integral $$\displaystyle\int x^2 \sin(x) dx$$?
$$\displaystyle x^2 \cos(x) - 2x \sin(x) + 2 \cos(x) + C$$
$$\displaystyle -x^2 \cos(x) + 2x \sin(x) + C$$
$$\displaystyle (2 - x^2) \cos(x) + 2x \sin(x) + C$$
$$\displaystyle (x^2 + 2) \sin(x) - 2x \cos(x) + C$$
$$\int$$x³sin(2x) dx =
-x³cos(2x)/2 + 3x²sin(2x)/4 + 3xcos(2x)/4 - 3sin(2x)/8 + C
-x³cos(2x)/2 + 3x²sin(2x)/4 - 3xcos(2x)/4 + 3sin(2x)/8 + C
x³sin(2x)/2 - 3x²cos(2x)/4 - 3xsin(2x)/4 + 3cos(2x)/8 + C
-x³cos(2x)/2 + 3x²sin(2x)/4 + 3xcos(2x)/4 - 3sin(2x)/4 + C
To evaluate the integral $$\int$$x^2 e^x dx, the method of integration by parts is applied twice. What is the integral that remains to be evaluated after the second application?
$$\int$$x^3 e^x dx
$$\int$$x^2 e^x dx
$$\int$$x e^x dx
$$\int$$e^x dx
Under which of the following conditions is it necessary to apply integration by parts a second time?
When the resulting integrand contains a linear function
When using integration by parts the first time still results in an integral that is a product of two functions
When the integrand is a polynomial function
When the resulting integrand is a polynomial
Which of the following is the correct initial step to evaluate $$\int x \ln x dx$$ using integration by parts?
Choose $$u = x$$ and $$dv = \ln x dx$$
Choose $$u = \ln x$$ and $$dv = x dx$$
Differentiate $$x$$ and integrate $$\ln x$$
Use substitution method
The tabular method for repeated integration by parts is most efficient for evaluating which of the following integrals?
$$\int$$sec^3(x) dx
$$\int$$e^x tan(x) dx
$$\int$$x^2/(x^2+1) dx
$$\int$$x^4 sin(x) dx