I.
$$\int (2*x+5) dx = x^2 + 5*x + C$$
II.
$$\int (2*x+5) dx = x^2 + \frac{5*x^2}{2} + C$$
III.
$$\int (2*x+5) dx = x*(2*x+5) + C$$
Which of the statements above is true?
The power rule, polynomial/linear/power functions, and basic algebraic integration.
60 questions
I.
$$\int (2*x+5) dx = x^2 + 5*x + C$$
II.
$$\int (2*x+5) dx = x^2 + \frac{5*x^2}{2} + C$$
III.
$$\int (2*x+5) dx = x*(2*x+5) + C$$
Which of the statements above is true?
I only
I and II
III only
II only
Which of the following is the general antiderivative of $$3x^2 + 2x$$?
$$x^3+x^2+C$$.
$$x^3+2*x^2+1$$ (missing the constant of integration format).
$$\frac{x^3}{3}+\frac{x^2}{2}+C$$.
$$3*x^3+2*x^2+C$$.
I. $$\int (3*x^2+4*x) dx = x^3 + 2*x^2 + C$$
II. $$\int (3*x^2+4*x) dx = \frac{3*x^3}{3} + \frac{4*x^2}{2} + C$$
III. $$\int (3*x^2+4*x) dx = \frac{3*x^3}{2} + 2*x^2 + C$$
Which of the statements above are correct?
I, II, and III
I and II only
I only
II and III only
What is an antiderivative of the function $$f(x) = 3x^2 + 2x - 5$$?
$$x^3 + x^2 - 5*x$$
$$x^2 + 2*x - 5 + C$$
$$3*x^3 + x^2 - 5*x + C$$
$$x^3 + x^2 - 5*x + C$$
Which of the following is the general antiderivative of $$2x$$?
$$2*x^2+C$$ due to an extra multiplication factor.
$$x^2$$ (omitting the constant of integration).
$$2*x+C$$.
$$x^2+C$$.
Which of the following statements is correct?
I. $$\int 3*x^3 dx = \frac{3*x^4}{4} + C$$
II. $$\int 3*x^3 dx = \frac{3*x^4}{4}$$
III. $$\int 3*x^3 dx = \frac{x^4}{4} + C$$
III only
I only
II only
II and III
I. $$\int x^{1/2} dx = \frac{2}{3} x^{3/2} + C$$
II. $$\int x^{1/2} dx = \frac{x^{3/2}}{3/2} + C$$
III. $$\int x^{1/2} dx = \frac{3}{2} x^{3/2} + C$$
Which of the statements above correctly gives the indefinite integral of $$x^{1/2}$$?
I only
I, II, and III
I and II only
II and III only
Evaluate the integral: $$\int \left(\frac{3}{\sqrt{x}} - 2x^{1/3}\right)dx$$
$$6*\sqrt{x}-\frac{3}{4}x^{4/3}+C$$
$$3*\sqrt{x}-\frac{3}{2}x^{4/3}+C$$
$$6*\sqrt{x}+\frac{3}{2}x^{4/3}+C$$
$$6*\sqrt{x}-\frac{3}{2}x^{4/3}+C$$
Which of the following properties of integrals states that the integral of a constant multiple of a function is the constant multiple of the integral of the function?
Power rule
Linearity property
Mean Value Theorem
Substitution property
All of the following statements regarding the integration of a constant multiple of a function are true EXCEPT:
When integrating a constant times a function, you must square the constant, so that $$\int k * f(x) dx = k^2 * \int f(x) dx.$$
For example, $$\int 3*x^2 dx = 3*\frac{x^3}{3} + C = x^3 + C.$$
The constant multiple rule simplifies integration by permitting direct use of the power rule once the constant is factored out.
The constant multiple rule allows you to factor out the constant: $$\int k * f(x) dx = k * \int f(x) dx.$$
Using the graph provided, evaluate the definite integral $$\int_{1}^{3} (2*x+3) dx$$.
$$14$$
$$12$$
$$16$$
$$10$$
What is the value of $$\int_{0}^{2} (x+3) dx$$?
$$8 + C$$
$$8$$
$$7$$
$$10$$
Evaluate $$\int_{2}^{5} 3(x-2) dx$$.
$$14$$
$$12$$
$$15$$
$$13.5$$
Evaluate $$\int_{1}^{4} (2x+1) dx$$.
$$16$$
$$20$$
$$2$$
$$18$$
What is the value of the definite integral of the function f(x) = 2x on the interval [0, 4]?
4
14
8
12
Evaluate $$\int_{0}^{4} (x^2+2x) dx$$.
37
40
36
$$\frac{112}{3}$$
Which of the following statements is true regarding the definite integral $$\int_{0}^{2}(x^3+1) dx$$?
The definite integral equals $$10$$ from misapplying the evaluation at the endpoints.
The definite integral equals $$6$$.
The definite integral equals $$4$$ by using an incorrect antiderivative.
The definite integral equals $$8$$ due to a miscalculation of the integrated terms.
Evaluate the definite integral $$\int_{-1}^{2} (3x^2 - 2x + 1) dx$$.
$$5$$
$$12$$
$$6$$
$$9$$
Evaluate the definite integral $$\int_{0}^{2} (3x^2+2x+1) dx$$.
$$10$$
$$12$$
$$16$$
$$14$$
Evaluate $$\int_{1}^{2} \left(4x^3 - x\right) dx$$
$$27$$
$$\frac{27}{2}$$
$$57$$
$$36$$
$$\displaystyle\int_{-2}^{2} \left(x^2 - 4\right) dx = $$
$$\displaystyle-\frac{32}{3}$$
$$\displaystyle\frac{32}{3}$$
$$0$$
$$\displaystyle-\frac{16}{3}$$
Evaluate $\displaystyle\int_{1}^{3} \left(x^3 - 3x^2 + 2x\right) dx$.
$$2$$
$$\displaystyle\frac{9}{4}$$
$$0$$
$$8$$
Evaluate the definite integral $$\int_{0}^{1} (x+2)^3 dx$$.
$$\frac{65}{3}$$
$$\frac{49}{4}$$
$$\frac{49}{3}$$
$$\frac{65}{4}$$
Which of the following represents the correct evaluation of the definite integral $$\int_{1}^{4}\frac{x}{\sqrt{x}} dx$$?
Value = $$\frac{14}{3}+C$$, erroneously including a constant in a definite integral.
Value = $$\frac{2}{3}(4^{3/2}-1^{3/2})=\frac{14}{3}$$
Value = $$\frac{7}{3}$$, by misinterpreting the exponents.
Value = Approximately 4.67, lacking proper algebraic form.
Compute the definite integral $$\int_{0}^{9} 3*\sqrt{x} dx$$ using the Fundamental Theorem of Calculus.
An error might yield $$36$$ after misinterpreting the exponent.
$$54$$
$$81$$
$$27$$
What is the value of $\displaystyle \int_{1}^{4} t^{-3/2} dt$?
$$-1$$
$$-\frac{1}{2}$$
$$1$$
$$-\frac{7}{8}$$
Evaluate $$\int_{1}^{4} 3\sqrt{x} dx$$.
$$14$$
$$12$$
$$16$$
$$10$$
Evaluate $$\int_{0}^{3} \sqrt{x} dx$$.
$$2\sqrt{3}$$
$$\frac{2}{3}\sqrt{3}$$
$$\frac{3}{2}\sqrt{3}$$
$$3\sqrt{2}$$
What is the value of $$\displaystyle\int_{0}^{2} \left(2x^4 - 6x^2 + 4\right) dx$$?
$$8$$
$$\displaystyle\frac{24}{5}$$
$$\displaystyle\frac{48}{5}$$
$$4$$
I. $$\int_{1}^{2} \frac{2}{x^2} dx = 1$$
II. $$\int_{1}^{2} \frac{2}{x^2} dx = -1$$
III. $$\int_{1}^{2} \frac{2}{x^2} dx = 2^{-2} - 1^{-2}$$
Which of the statements above is correct?
I and II
II only
I only
III only
If the derivative of a function `f` is given by `f'(x) = 2x`, which of the following is the general expression for `f(x)`?
2x + C
1/x + C
2x - C
x^2 + C
What is the general antiderivative of a function $f$ defined by $f(x) = 6x - \frac{4}{x^2}$?
3x² + 4/x²
3x² + 4/x + C
3x² - 4/x
3x² - 4/x²
Which of the following is the general antiderivative of `f(x) = 2x³ - 5/x²`?
(1/2)x^4 - 5/x + C
(2/3)x^4 - 5/x²
(2/3)x^4 - 5/x
(1/2)x^4 - 5/x²
Evaluate $$\int (4x^3 - 2x + 1) dx$$.
$$x^4 - x^2 + x + C$$
$$x^4 - 2*x^2 + x + C$$
$$x^4 + x^2 + x + C$$
$$4*x^4 - 2*x^2 + x + C$$
Determine an antiderivative of $$f(x)=3*x^2+2*x-5$$. Provide the indefinite integral including the constant of integration.
$$x^3+x^2-5*x+C$$
$$3*x^3+2*x^2-5*x+C$$
$$x^3+ x^2-5+C$$
$$x^3+x^2-5*x$$ (missing the constant)
Which of the following is equal to $$\int 4(2x+3) dx$$?
Antiderivative = $$2*x^2+12*x+C$$
Antiderivative = $$4*x^2+12*x+C$$
Antiderivative = $$8*x^2+12*x+C$$
Antiderivative = $$4*x^2+12*x$$ (missing C)
$$\int (3x^2 + 2x + 1) dx =$$
x³ + x² + x + C
x³ + x + 1
x² + x + 1
x³ + x² + x
Evaluate $$\int$$(5cos(x) + 2x) dx.
5sin(x) + x
5sin(x) + 2x²
5cos(x) + x²
5sin(x) + x² + C
Which of the following is equivalent to $$\int 3(x-2)^4 dx$$?
$$\frac{3*(x-2)^5}{4}+C$$ due to an incorrect exponent adjustment.
$$3*\frac{(x-2)^4}{4}+C$$, which misapplies the constant factor.
$$\frac{3*(x-2)^5}{5}+C$$.
$$\frac{(x-2)^5}{5}+C$$, missing the coefficient 3 in the result.
Evaluate $$\int$$(3x² + 2√(x)) dx.
3x^2 + 2x
x^3 + x^(3/2)
3x^2 + (4/3)x^(3/2)
x^3 + (4/3)x^(3/2) + C
If $$\int$$f(x)dx = 3x⁵ - 2x³ + 7x + C and $$\int$$g(x)dx = 4x⁴ - 6x² + 2 + C, which of the following is $$\int \left(2f(x) - 3g(x)\right)$$dx?
6x⁵ - 4x³ + 14x - 12x⁴ + 18x² - 6 + C
2(3x⁵ - 2x³ + 7x) - 3(4x⁴ - 6x² + 2) + C
6x⁵ - 4x³ + 14x - 12x⁴ + 18x² - 6
15x⁴ - 10x² + 35 - 12x³ + 18x + 6 + C
Rank the integrals $$\int x^n dx$$ for n = 0, 1, 2, and 3 in ascending order according to the exponent of x in their resulting antiderivatives.
$$\int x^3 dx$$, $$\int x^2 dx$$, $$\int x^1 dx$$, $$\int x^0 dx$$
$$\int x^0 dx$$, $$\int x^1 dx$$, $$\int x^2 dx$$, $$\int x^3 dx$$
$$\int x^1 dx$$, $$\int x^0 dx$$, $$\int x^3 dx$$, $$\int x^2 dx$$
$$\int x^0 dx$$, $$\int x^2 dx$$, $$\int x^1 dx$$, $$\int x^3 dx$$
Which of the following is an example of applying the power rule to find the antiderivative of $$3*x^2$$?
Antiderivative: $$\frac{x^3}{3}+C$$
Antiderivative: $$x^3+C$$
Antiderivative: $$\frac{3*x^3}{2}$$ (missing C)
Antiderivative: $$\frac{3*x^3}{2}+C$$
All of the following statements about the power rule for antiderivatives are true EXCEPT:
For a constant multiple, the constant can be factored out: $$\int k*x^n dx = k*\int x^n dx$$.
If $$n\neq -1$$, then $$\int x^n dx = \frac{x^{n+1}}{n+1}+C$$.
The restriction $$n\neq -1$$ is necessary because the power rule does not apply when $$n = -1$$.
When applying the power rule, $$\int k*x^n dx$$ equals $$k*(n+1)*x^{n+1}+C$$.
Which of the following represents the general antiderivative of $$5x^4$$?
Antiderivative = $$5*x^5+C$$
Antiderivative = $$\frac{x^5}{5}+C$$
Antiderivative = $$x^5+C$$
Antiderivative = $$x^5$$ (missing C)
Which of the following is the general antiderivative of $$x^3$$?
$$3*x^3+C$$.
$$\frac{x^4}{4}+C$$.
$$4*x^3+C$$.
$$x^4+C$$.
Which of the following best describes the power rule for antidifferentiation?
The rule used to find the derivative of a polynomial.
The rule used to find the antiderivative of a power function.
The rule used to find the derivative of an exponential function.
The rule used to find the derivative of a constant.
I. $$\int x^{-3} dx = -\frac{1}{2*x^2} + C$$
II. $$\int x^{-3} dx = -\frac{1}{3*x^3} + C$$
III. $$\int x^{-3} dx = \frac{1}{2*x^2} + C$$
Which of the following statements is true?
I only
II only
I and III
III only
What is $$\int 4x^{1/3} dx$$?
$$\frac{4*x^{4/3}}{3} + C$$
$$3*x^{1/3} + C$$
$$\frac{3}{4}* x^{4/3} + C$$
$$3*x^{4/3} + C$$
Evaluate the indefinite integral $$\int x^3 dx$$.
$$\frac{1}{4}x^3 + C$$
$$\frac{x^4}{4} + C$$
$$3*x^2 + C$$
$$\frac{x^4}{3} + C$$
What is $$\int x^{-2} dx$$?
$$-\frac{1}{x} + C$$
$$\frac{1}{x} + C$$
$$-\frac{1}{x^2} + C$$
$$\frac{-1}{2}*x^{-2} + C$$
Which of the following is equal to $$\int \frac{1}{\sqrt{x}} dx$$?
$$\frac{\sqrt{x}}{2}+C$$, which incorrectly divides the result.
$$2*\sqrt{x}+C$$.
$$\frac{1}{2}*\ln\lvert x\rvert+C$$, which confuses the integration with a logarithmic form.
$$\sqrt{x}+C$$, missing the necessary constant multiplier.
What is $$\int 2x dx$$?
$$x^2$$
$$2*x^2 + C$$
$$x^2 + C$$
$$2*x + C$$
$$\int x^{\frac{3}{2}} dx =$$
Antiderivative = $$\frac{2}{5}*x^{\frac{5}{2}}+C$$
Antiderivative = $$\frac{3}{5}*x^{\frac{5}{2}}+C$$
Antiderivative = $$\frac{2}{3}*x^{\frac{5}{2}}+C$$
Antiderivative = $$\frac{2}{5}*x^{\frac{3}{2}}+C$$
Which of the following is an antiderivative of $$\sqrt{x}$$?
$$\frac{2}{3}* x^{1/2} + C$$
$$\frac{2}{3}* x^{3/2} + C$$
$$\frac{1}{3}* x^{3/2} + C$$
$$\frac{3}{2}* x^{3/2} + C$$
$$\displaystyle \int \frac{1}{t\sqrt{t}} dt = $$
$$\displaystyle -2t^{-1/2} + C$$
$$\displaystyle 2t^{1/2} \ln t + C$$
$$\displaystyle -\frac{2}{5}t^{-5/2} + C$$
$$\displaystyle -\frac{3}{2}t^{-5/2} + C$$
$$\frac{1}{3}\int \sqrt{u} du = $$
(2u^(2/3))/9+C
(2u^(3/2))/9+C
(2u^(3/2))/9
(2u^(3/2))/3 + C
Which of the following lists the correct sequence of steps for evaluating an indefinite integral using the power rule?
Simplify the integrand, Rewrite radical expressions as exponents, Apply the power rule to integrate, Add the constant of integration
Rewrite radical expressions as exponents, Apply the power rule to integrate, Simplify the integrand, Add the constant of integration
Add the constant of integration, Rewrite radical expressions as exponents, Simplify the integrand, Apply the power rule to integrate
Rewrite radical expressions as exponents, Simplify the integrand, Apply the power rule to integrate, Add the constant of integration
Which property of definite integrals states that the integral of a sum of functions is equal to the sum of the integrals of the individual functions?
Integration by Parts
Fundamental Theorem of Calculus
Substitution property
Linearity property
Which property of definite integrals is used to rewrite the integral of a sum of two functions as the sum of their individual integrals?
Linearity property
Mean Value Theorem
Fundamental Theorem of Calculus
Substitution property