Find the general antiderivative of $$h(x)=\frac{3x^2}{\sqrt{2x^3+5}}$$.
Choosing u, changing limits of integration, and substitution technique (incl. trig substitution).
87 questions
AP Learning Target(s): LT 9: Integrating Using Substitution
Find the general antiderivative of $$h(x)=\frac{3x^2}{\sqrt{2x^3+5}}$$.
$$\sqrt{2*x^3+5}-C$$
$$\frac{3}{\sqrt{2*x^3+5}}+C$$
$$\sqrt{2*x^3+5}+C$$
$$\frac{1}{2}\sqrt{2*x^3+5}+C$$
What is $$\int (x - 4)^{10} dx$$?
$$\frac{(x-4)^{10}}{10} + C$$
$$\frac{(x-4)^{11}}{10} + C$$
$$\frac{(x-4)^{11}}{11} + C$$
$$(x-4)^{11} + C$$
Evaluate the integral: $$\int \left(\sqrt{x} + \frac{1}{x+3} - \frac{x}{\sqrt{x+1}}\right) dx$$
$$\frac{2}{3}x^{3/2}+\ln|x+3|-\frac{2}{3}(x+1)^{3/2}+2*(x+1)^{1/2}-C$$
$$\frac{2}{3}x^{3/2}+\ln|x+3|+\frac{2}{3}(x+1)^{3/2}-2*(x+1)^{1/2}+C$$
$$\frac{2}{3}x^{3/2}+\ln|x+3|-\frac{2}{3}(x+1)^{3/2}+C$$
$$\frac{2}{3}x^{3/2}+\ln|x+3|-\frac{2}{3}(x+1)^{3/2}+2*(x+1)^{1/2}+C$$
Which of the following substitutions is the most appropriate for evaluating the integral $$\int$$(ln x)²/x dx?
u = (ln x)²
u = ln x
u = 1/x
u = x ln x
Which of the following substitutions is most appropriate for evaluating the integral $$\int$$√(4-x²)dx?
u = 4-x²
x = 2sin(θ)
x = 2tan(θ)
u = √(4-x²)
When using integration by parts to evaluate the integral $$\int$$(ln(x))^2 dx, which of the following is the most effective choice for u and dv?
u = ln(x)^2, dv = dx
u = 1, dv = ln(x)^2 dx
u = x, dv = ln(x)^2 dx
u = ln(x), dv = ln(x) dx
$$Which of the following is the most appropriate choice for `u` to evaluate the integral `\int_{0}^{2} x\sqrt{3x^2 + 7} dx` by substitution?$$
u = x
u = sqrt(3x^2 + 7)
u = 3x
u = 3x^2 + 7
Evaluate the definite integral: $$\displaystyle\int_{0}^{\ln 5} \frac{e^x + 3}{e^x + 3x} dx$$
$$e^{\ln 5} + 3\ln 5$$
$$\ln(5 + 3\ln 5)$$
$$5 + 3\ln 5 - 1$$
$$\ln 5 + 3$$
If the substitution u = ln x is used to evaluate the definite integral $\displaystyle\int_{1}$^₄ (ln x)/x dx, which of the following statements is correct?
The new integral is $\displaystyle\int_{0}$^ln 4 u du with value (ln 4)²/2
The new integral is $\displaystyle\int_{0}$^ln 4 u du
The substitution doesn't simplify this integral
The new integral is $$\int$$ln 1^ln 4 u du
$$\displaystyle\int_{1}^{2} \frac{3x^2 + 2}{x^3 + 2x} dx = $$
$$\ln(3x^2 + 2) - \ln(x^3 + 2x)$$
$$\ln 3$$
$$\ln 4$$
$$\frac{12}{3}$$
What is the value of $$\displaystyle\int_{0}^{1} \frac{2x}{x^2 + 1} dx$$?
$$\displaystyle\frac{1}{2}$$
$$\ln(2)$$
$$1$$
$$\displaystyle\frac{\pi}{4}$$
Evaluate the definite integral $$\displaystyle\int_{0}^{\ln 3} \frac{3e^x - 2}{3e^x - 2x - 1} dx$$.
$$8 - 2\ln 3 - 2$$
$$\ln(4 - \ln 3)$$
$$\ln(8 - 2\ln 3)$$
$$\ln(4 - \ln 3) - \ln 2$$
What is the value of the definite integral $$\displaystyle \int_{0}^{1} x\sqrt{1 + 8x^2} dx$$?
$$\displaystyle \frac{52}{3}$$
$$\displaystyle \frac{9}{8}$$
$$\displaystyle \frac{1}{24}$$
$$\displaystyle \frac{13}{12}$$
What is the value of the definite integral $$ \displaystyle\int_{0}^{\ln 2} \frac{e^x + 2}{e^x + 2x + 1} dx $$?
$$2\ln 2 + 1$$
$$3 + 2\ln 2 - 2$$
$$\ln\left(\frac{3 + 2\ln 2}{2}\right)$$
$$\ln(2 + 2\ln 2) - \ln 2$$
Evaluate $\displaystyle\int_{0}^{\ln 4} \frac{2e^x + 1}{2e^x + x} dx$.
$$\ln(8 + \ln 4) + \ln 2$$
$$2\ln 4 + 1$$
$$\ln\left(\frac{8 + \ln 4}{2}\right)$$
$$8 + \ln 4 - 2$$
Evaluate $$\int_{1}^{e}\frac{(\ln x)^5}{x} dx$$.
1/6
1/2
1/12
1/24
Evaluate $$\int_{0}^{\pi} \sin^3(x) \cos(x) dx$$.
0.5
0.25
0
1
Evaluate the definite integral $$\int_{1}^{3} 2*(x-1) dx$$ using u-substitution.
Approximately $$3.5$$ from misapplied substitution.
$$8$$
$$4$$
$$2$$
Evaluate the definite integral $$\int_{1}^{5} \frac{2*x}{\sqrt{x^2+1}} dx$$ using a substitution. Which of the following is correct?
$$2*\sqrt{26}+2*\sqrt{2}$$
$$\sqrt{26}-\sqrt{2}$$
$$2*\sqrt{26}-2*\sqrt{2}$$
$$\frac{\sqrt{26}-\sqrt{2}}{2}$$
Evaluate the definite integral $$\int_{2}^{6} (x-4)^5 dx$$.
$$\frac{2^6+(-2)^6}{6}$$
$$\frac{(6-4)^6 - (2-4)^6}{6}$$
$$0$$
$$\frac{(6-4)^6+ (2-4)^6}{6}$$
Evaluate the definite integral $$\int_{0}^{4} \frac{x}{\sqrt{x+2}} dx$$. Use an appropriate substitution and simplify your answer.
$$\frac{8*\sqrt{2}}{3}$$
$$\frac{4*\sqrt{6}-4*\sqrt{2}}{3}$$
$$\frac{8*\sqrt{6}}{3}$$
$$\frac{2*\sqrt{6}}{3}$$
Evaluate $$\int_{1}^{5} (2x+3)^4 dx$$.
$$\frac{1}{10}(13^5-5^5)$$
$$\frac{1}{2}(13^5-5^5)$$
$$\frac{1}{5}(13^5-5^5)$$
$$\frac{1}{10}(13^5+5^5)$$
What is the value of $$\int_{0}^{1}(2x+3)^5 dx$$?
Value = $$\frac{5^6-3^6}{12}$$
Value = $$\frac{5^6-3^6}{10}$$
Value = $$\frac{5^6-3^6}{6}$$
Value = $$\frac{5^5-3^5}{12}$$
Evaluate $$\int_{4}^{5} (x-4)^{10} dx$$.
$$\frac{1}{11}$$
$$\frac{1}{10}$$
$$\frac{1}{9}$$
$$\frac{1}{12}$$
Evaluate $$\int_{2}^{6} (x-2)^3 dx$$.
128
16
64
32
Using the substitution $$u = x^2 - 3$$, the definite integral $$\displaystyle \int_{1}^{4} x(x^2 - 3)^5 dx$$ is equivalent to which of the following?
$$\displaystyle \int_{-1}^{4} u^5 du$$
$$\displaystyle \frac{1}{2}\int_{-2}^{13} u^5 du$$
$$\displaystyle \int_{-2}^{13} u^5 du$$
$$\displaystyle 2\int_{-2}^{13} u^5 du$$
What is the value of the definite integral $$\int_{0}^{\}pi \frac{e^x - 1}{e^x - x} dx$$?
$$\pi - \ln \pi$$
$$\ln(e^\pi - \pi)$$
$$e^\pi - \pi - 1$$
$$\ln(e^\pi - \pi) - 1$$
What is the value of $\displaystyle \int_{0}^{4} \frac{x}{\sqrt{x^2 + 9}} dx$?
$$-\dfrac{2}{15}$$
$$1$$
$$2$$
$$-2$$
What is the value of $$\displaystyle\int_{0}^{\ln(3/2)} \frac{2e^x - 1}{2e^x - x} dx$$?
$$\ln 3 - \ln 2 - \frac{\ln(3/2)}{2}$$
$$2\ln 3 - \ln 2$$
$$3 - \ln(3/2) - 2$$
$$\ln\left(\frac{3 - \ln(3/2)}{2}\right)$$
Using the substitution u = x², what is the value of the definite integral $\displaystyle\int_{0}$^₁ xe^(x²) dx?
e/2
(e-1)/2
(e²-1)/4
e-1
Using the substitution u = x² + 1, which of the following definite integrals is equivalent to $\displaystyle\int_{1}$^₄ (x / (x² + 1)²) dx?
$\displaystyle\int_{2}$^₁₇ 1/(2u²)du
$\displaystyle\int_{2}$^₁₇ -1/(2u²)du
$\displaystyle\int_{2}$^₁₇ 1/(2u²)du
$\displaystyle\int_{2}$^₁₇ 1/(4u²)du
What is the value of $\displaystyle\int_{1}^{4}$ xe^(x²-1) dx?
(e¹⁶ - 1)/2
(e¹⁵ - e)/2
e¹⁵ - e
(e¹⁵ - 1)/2
All of the following statements comparing direct integration and u-substitution are true EXCEPT:
U-substitution is particularly useful when the integrand is a composite function and the derivative of the inner function is present.
The choice between direct integration and u-substitution depends on the structure of the integrand, with u-substitution simplifying composite functions.
Direct integration is most efficient when the integrand is already in a form that directly permits the use of the power rule.
Direct integration using the power rule can always be applied even when the integrand is a composite function without the need for u-substitution.
Which of the following integrals is most efficiently evaluated using the substitution $u = x^2 + 1$?
$$\int$$(x² + 1)⁻¹ dx
$$\int$$x⁴√(x² + 1) dx
$$\int$$x³(x² + 1)² dx
$$\int$$x(x² + 1)⁵ dx
I.
$$\int_{0}^{4} \sqrt{x+1} dx = \frac{2}{3}(5^{3/2}-1)$$
II.
$$\int_{0}^{4} \sqrt{x+1} dx = \frac{1}{3}(5^{3/2}-1^{3/2})$$
III.
$$\int_{0}^{4} \sqrt{x+1} dx = \frac{2}{3}(5^{3/2}+1)$$
Which of the preceding statements is true?
II only
II and III
I only
III only
The growth rate of a population, $P$, is described by the differential equation $\frac{dP}{dt} = 0.8P$. If the population at time $t=0$ is 50, what is the population at time $t=5$?
$$40e^4$$
$$50e^4$$
$$50e^5$$
$$50e^3$$
If u = 3x + 7, which of the following expresses the differential du in terms of the differential dx?
du = 7 dx
du = 10 dx
du = 3x dx
du = 3 dx
Evaluate the integral $$\displaystyle\int \frac{(3x^{3/4} + 2)^5}{4x^{1/4}} dx$$.
$$\displaystyle\frac{(3x^{3/4} + 2)^6}{24} + C$$
$$\displaystyle\frac{(3x^{3/4} + 2)^6}{18} + C$$
$$\displaystyle\frac{(3x^{3/4} + 2)^6}{54} + C$$
$$\displaystyle\frac{5(3x^{3/4} + 2)^4}{3x^{1/4}} + C$$
Evaluate $$\int$$(ln(x))^2/x dx.
(ln(x))^3/3 + C
(ln(x))^3 - 2(ln(x))^2 + C
x(ln(x))^2 - 2x·ln(x) + 2x + C
(ln(x))^2/2 + C
Evaluate $$\int (x-4)^{10} dx$$.
$$\frac{(x-4)^{11}}{11}+C$$.
$$\frac{(x-4)^{10}}{10}+C$$.
$$\frac{(x-4)^{11}}{10}+C$$.
$$(x-4)^{11}+C$$ (omitting the necessary division).
Evaluate $$\int (x - 4)^{10} dx$$.
$$\frac{(x-2)^{11}}{11}+C$$
$$\frac{(x-4)^{11}}{10}+C$$
$$\frac{(x-4)^{11}}{11}+C$$
$$\frac{(x-4)^{10}}{10}+C$$
I.
$$\int (3x-1)^4 dx = \frac{(3x-1)^5}{15} + C$$
II.
$$\int (3x-1)^4 dx = \frac{(3x-1)^5}{5} + C$$
III.
$$\int (3x-1)^4 dx = \frac{(3x-1)^5}{15}$$
Which of the statements above is true?
II and III
I only
II only
III only
Evaluate $$\displaystyle\int \frac{(2x^{1/3} + 1)^4}{3x^{2/3}} dx$$
$$\displaystyle\frac{4(2x^{1/3} + 1)^3}{x^{2/3}} + C$$
$$\displaystyle\frac{(2x^{1/3} + 1)^5}{10} + C$$
$$\displaystyle\frac{(2x^{1/3} + 1)^5}{15} + C$$
$$\displaystyle\frac{2(2x^{1/3} + 1)^5}{15} + C$$
Which of the following integrals requires u-substitution for evaluation?
$$\int$$ (4x+6)^2 dx
$$\int$$ 4x+6^2 dx
$$\int$$ 4x^2+6 dx
$$\int$$ 4x+6 dx
Which of the following integrals can be evaluated without using the method of u-substitution?
$$\int$$ (4x^2+6x)^(9/2) dx
$$\int$$ 4x+6 dx
$$\int$$ (4x+6)^2 dx
$$\int$$ sqrt(4x^2+6) dx
Evaluate $$\displaystyle\int (4\sqrt{x} - 1)^3 \cdot \frac{1}{\sqrt{x}} dx$$
$$\displaystyle\frac{(4\sqrt{x} - 1)^4}{8} + C$$
$$\displaystyle\frac{3(4\sqrt{x} - 1)^2}{\sqrt{x}} + C$$
$$\displaystyle\frac{(4\sqrt{x} - 1)^4}{4} + C$$
$$\displaystyle\frac{(4\sqrt{x} - 1)^4}{16} + C$$
Which of the following is equivalent to $$\displaystyle \int x^2(x^3 + 5)^6 dx?$$
$$\displaystyle \frac{1}{3}x^3\left(\displaystyle \frac{1}{4}x^4 + 5x\right)^6 + C$$
$$\displaystyle \frac{1}{3}(x^3 + 5)^6 + C$$
$$\displaystyle \frac{1}{21}(x^3 + 5)^7 + C$$
$$\displaystyle \frac{1}{7}(x^3 + 5)^7 + C$$
$$\int$$x^3(x^4+2)^6 dx =
(1/28)(x^4+2)^7 + C
(1/24)x^4(x^4+2)^6 + C
(1/4)(x^4+2)^7 + C
(1/7)(x^4+2)^7 + C
If $$\int$$f(g(x))g'(x)dx = F(g(x)) + C, then $$\int$$sin(3x²)·6x dx =
-(1/6)cos(3x²) + C
-cos(3x²) + C
-(1/3)cos(3x²) + C
-2cos(3x²) + C
Which of the following integrals is equivalent to $$\int$$sqrt(3x+7)dx after an appropriate u-substitution is made?
1/3$$\int$$sqrt(u) du
$$\int$$sqrt(u/3) du
$$\int$$sqrt(u) dx
$$\int$$sqrt(u) du
Which of the following integration techniques involves simplifying an integral by making a substitution?
Partial fractions
Integration by parts
u-substitution
Trigonometric substitution
Evaluate $$\int_{1}^{2} \frac{\sqrt{x^2-1}}{x} dx$$.
$$$ \ln(2 + \sqrt{3}) - \ln(1 + 0) $$$
$$$ \sqrt{3} - \ln(2 + \sqrt{3}) $$$
$$$ \sqrt{3} - 1 $$$
$$$ \frac{\sqrt{3}}{2} $$$
Use u-substitution to find the antiderivative of $$\cos(x)(\sin(x))^3$$.
$$\sin(x)*(\sin(x))^3 + C$$
$$\frac{(\cos(x))^4}{4} + C$$
$$-\frac{(\sin(x))^4}{4} + C$$
$$\frac{(\sin(x))^4}{4} + C$$
Evaluate the integral $$\int (x-4)^{10} dx$$.
Antiderivative = $$\frac{(x-4)^{11}}{11}+C$$
Antiderivative = $$\frac{(x-4)^{11}}{10}+C$$
Antiderivative = $$\frac{(x-4)^{10}}{10}+C$$
Antiderivative = $$(x-4)^{10}+C$$
Evaluate $$\int \frac{(x^{1/3} - 4)^5}{6x^{2/3}} dx$$
$$\dfrac{(x^{1/3} - 4)^6}{6} + C$$
$$\dfrac{5(x^{1/3} - 4)^4}{2} + C$$
$$3(x^{1/3} - 4)^6 + C$$
$$\dfrac{(x^{1/3} - 4)^6}{12} + C$$
Evaluate $$\displaystyle\int (x^{1/2} - 3)^6 \cdot \frac{1}{\sqrt{x}} dx$$
$$\displaystyle\frac{2(x^{1/2} - 3)^7}{7} + C$$
$$\displaystyle 2(x^{1/2} - 3)^7 + C$$
$$\displaystyle\frac{(x^{1/2} - 3)^7}{7} + C$$
$$\displaystyle\frac{6(x^{1/2} - 3)^5}{\sqrt{x}} + C$$
Evaluate the integral $$\displaystyle\int \frac{(3x^{2/3} - 5)^4}{x^{1/3}} dx$$.
$$\displaystyle\frac{(3x^{2/3} - 5)^5}{10} + C$$
$$\displaystyle\frac{2(3x^{2/3} - 5)^5}{15} + C$$
$$\displaystyle\frac{(3x^{2/3} - 5)^5}{6} + C$$
$$\displaystyle\frac{4(3x^{2/3} - 5)^3}{x^{1/3}} + C$$
Evaluate the integral $$\displaystyle\int \frac{(2x^{1/4} + 1)^3}{x^{3/4}} dx$$.
$$\displaystyle\frac{(2x^{1/4} + 1)^4}{2} + C$$
$$\displaystyle\frac{(2x^{1/4} + 1)^4}{8} + C$$
$$\displaystyle 2(2x^{1/4} + 1)^4 + C$$
$$\displaystyle\frac{4(2x^{1/4} + 1)^3}{x^{3/4}} + C$$
$$\displaystyle \int \frac{x^2 + 1}{(x^3 + 3x - 5)^3} dx =$$
$$\displaystyle -\frac{1}{6} \cdot \frac{1}{(3x^2 + 3)^2} + C$$
$$\displaystyle -\frac{1}{6} \cdot \frac{1}{(x^3 + 3x - 5)^2} + C$$
$$\displaystyle -\frac{3}{2} \cdot \frac{1}{(x^3 + 3x - 5)^2} + C$$
$$\displaystyle -\frac{3}{2} \cdot \frac{1}{(3x^2 + 3)^2} + C$$
Which of the following techniques is most appropriate for evaluating $$\int_{0}^{2} x \sqrt{3x^2 + 7} dx$$?
U-substitution
Completing the square
Chain rule
Product rule
Which of the following is equivalent to $$\int 5\cos(5x) dx$$?
$$-\sin(5*x) + C$$
$$\frac{\sin(5*x)}{5} + C$$
$$\sin(5*x) + C$$
$$5*\sin(5*x) + C$$
Evaluate the integral $$\displaystyle\int \frac{(x^{3/5} + 1)^2}{5x^{2/5}} dx$$.
$$\displaystyle\frac{3(x^{3/5} + 1)^3}{5} + C$$
$$\displaystyle\frac{(x^{3/5} + 1)^3}{9} + C$$
$$\displaystyle\frac{2(x^{3/5} + 1)}{3x^{2/5}} + C$$
$$\displaystyle\frac{(x^{3/5} + 1)^3}{15} + C$$
$$\displaystyle\int \frac{(x^{2/5} - 2)^3}{x^{3/5}} dx =$$
$$\displaystyle\frac{3(x^{2/5} - 2)^2}{x^{3/5}} + C$$
$$\displaystyle\frac{5(x^{2/5} - 2)^4}{4} + C$$
$$\displaystyle\frac{(x^{2/5} - 2)^4}{4} + C$$
$$\displaystyle\frac{5(x^{2/5} - 2)^4}{8} + C$$
If the substitution u = 3x^2 + 7 is used to evaluate a definite integral with limits of integration from x=0 to x=2, what are the new limits of integration with respect to u?
7 and 19
0 and 7
0 and 19
2 and 7
All of the following statements about the method of u-substitution for integrating composite functions are true EXCEPT:
After integrating with respect to $$u$$, substituting back into the original variable yields the final antiderivative in terms of $$x$$.
Substituting into the integral gives a new integrand in terms of $$u$$ that can be integrated using the power rule.
For the substitution $$u=3*x+4$$, the derivative is $$du=3 dx$$, so that $$dx=\frac{du}{3}$$.
When applying u-substitution to an integral like $$\int (3*x+4)^5 dx$$, one may set $$u=3*x+4$$ without accounting for the derivative of the inner function.
All of the following statements about integration by u-substitution are true EXCEPT:
A key requirement for u-substitution is that the derivative $$du/dx$$ of the chosen substitution must be incorporated into the integral.
The effectiveness of u-substitution depends on choosing $$u$$ so that its derivative appears (or can be factored out) in the integrand.
In u-substitution, it is acceptable to choose a substitution $$u$$ even if its derivative does not appear in the integrand.
Once the substitution is made, the integral is rewritten in terms of $$u$$ and then integrated before substituting back to $$x$$.
All of the following statements regarding the method of integration by u-substitution are true EXCEPT:
The substitution process inherently involves computing $$du/dx$$ so that $$dx$$ can be correctly replaced in the integral.
After integrating with respect to $$u$$, one must substitute back to the original variable when the integral is indefinite.
In u-substitution, it is always appropriate to let $$u$$ equal the entire integrand regardless of whether its derivative appears in the integrand.
A proper substitution reduces the integrand to a form where the derivative $$du/dx$$ is present, simplifying the integration.
The integration method of u-substitution is most appropriate for which of the following situations?
When the function to integrate includes multiple variables
When the function to integrate is of the form g(f(x))
When the function to integrate includes complex numbers
When taking the derivative of a function with the form g(f(x))
Arrange the following integrals in increasing order of complexity, where complexity is determined by the number of steps required to evaluate them using u-substitution: $$\int \sqrt{x} dx$$, $$\int (3x+2)^4 \cdot 3 dx$$, $$\int (x^2+1)^5 \cdot 2x dx$$, and $$\int \frac{1}{\sqrt{1-x^2}} dx$$.
$$\int \sqrt{x} dx$$, $$\int (3*x+2)^4 * 3 dx$$, $$\int (x^2+1)^5 * 2*x dx$$, $$\int \frac{1}{\sqrt{1-x^2}} dx$$
$$\int \sqrt{x} dx$$, $$\int (x^2+1)^5 * 2*x dx$$, $$\int (3*x+2)^4 * 3 dx$$, $$\int \frac{1}{\sqrt{1-x^2}} dx$$
$$\int (3*x+2)^4 * 3 dx$$, $$\int \sqrt{x} dx$$, $$\int \frac{1}{\sqrt{1-x^2}} dx$$, $$\int (x^2+1)^5 * 2*x dx$$
$$\int \frac{1}{\sqrt{1-x^2}} dx$$, $$\int (x^2+1)^5 * 2*x dx$$, $$\int (3*x+2)^4 * 3 dx$$, $$\int \sqrt{x} dx$$
If the differentials du and dx are related by the equation du = 3 dx, which of the following correctly expresses dx in terms of du?
dx = du - 3
dx = 3 du
dx = du + 3
dx = 1/3 du
Which of the following is the most efficient substitution for evaluating the integral $$\int$$(tan(x)·sec^2(x))dx?
u = sin(x)/cos(x)
u = sec(x)
u = sec^2(x)
u = tan(x)
Which of the following is equivalent to $$\int \tan^2(x) \sec^2(x) dx$$?
⅓tan³(x) + C
tan(x) - x + C
½tan²(x) + C
tan(x)sec(x) + C
Which of the following substitutions is the most appropriate for evaluating the integral $$\int x^2 \sqrt{x^2 + 9} dx$$?
$$u = x^2$$ and then integration by parts
$$x = 3tan(\theta)$$
$$x = 3sinh(\theta)$$
$$u = x^2 + 9$$
Which of the following integrals requires both u-substitution and integration by parts to evaluate?
$$\int$$e^(3x)dx
$$\int$$(4x-1)/(x²-x)dx
$$\int$$x·ln(x²+1)dx
$$\int$$sin(5x)dx
If `u = 3x^2 + 7`, which of the following correctly expresses `dx` in terms of `du`?
dx = 6x du
dx = 1/6 du
dx = 1/6x du
du = 6/x dx
$$\displaystyle\int_{0}^{\ln 2} \frac{e^x + 1}{e^x + x} dx = $$
$$\ln(2 + \ln 2)$$
$$\ln 2 + \ln(2 + \ln 2)$$
$$2 + \ln 2$$
$$e^{\ln 2} - \ln 2$$
$$\displaystyle\int_{1}^{e} \frac{4x + 1}{2x^2 + x - 1} dx = $$
$$\frac{2e^2 + e - 1}{2}$$
$$2e^2 + e - 3$$
$$\ln(2e^2 + e - 1) + \ln 2$$
$$\ln\left(\frac{2e^2 + e - 1}{2}\right)$$
$$\displaystyle\int_{2}^{3} \frac{2x - 1}{x^2 - x + 1} dx = $$
$$\frac{7}{3} - 1$$
$$\ln 7 + \ln 3$$
$$\ln\left(\frac{7}{3}\right)$$
$$2\ln 3 - \ln 7$$
Consider the following statements regarding the definite integral $$\int_{1}^{5} (x-4)^{10} dx$$.
I. $$\int_{1}^{5} (x-4)^{10} dx = \frac{(5-4)^{11} - (1-4)^{11}}{11}$$
II. $$\int_{1}^{5} (x-4)^{10} dx = \frac{(1-4)^{11} + (5-4)^{11}}{11}$$
III. $$\int_{1}^{5} (x-4)^{10} dx = \frac{(5-4)^{11} + (1-4)^{11}}{11}$$
When evaluating the integral using the substitution $$u=x-4$$, the transformed integral is $$\int_{-3}^{1} u^{10} du$$, which equals $$\frac{u^{11}}{11}\Big|_{-3}^{1}$$. Which of the statements above correctly represents this result?
II only
III only
II and III
I only
When using u-substitution to evaluate a definite integral, what additional step must be performed?
Changing the bounds
Subtracting C from the integral
Adding u to the integral
Dividing the solution by x
Using the substitution $$u = \sin(x)$$, which of the following definite integrals is equivalent to $$\int_{0}^{\}pi \sin^3(x) \cos(x) dx$$?
$$\int_{0}^{\}pi u^3 du$$
$$\int_{0}^{0} u^3 du$$
$$\int_{0}^{0} u^3 du$$
$$\int_{-1}^{1} u^3 du$$
Which of the following best describes u-substitution?
An integration technique used to solve integrals involving complex numbers
An integration technique used to solve integrals of nested functions
An integration technique used to solve integrals involving multiple variables
A derivation technique used to solve derivatives of nested functions
If the integral $\displaystyle\int_{1}$^₄ x√(x²+9) dx is evaluated using the substitution u = x²+9, what are the new limits of integration?
u = 9 to u = 16
u = 10 to u = 16
u = 1 to u = 4
u = 10 to u = 25
$$\displaystyle \int \frac{e^x}{1 + e^x} dx =$$
$$x - \ln(1 + e^x) + C$$
$$e^x + x + C$$
$$\ln\left(\frac{1}{e^x} + 1\right) + C$$
$$\ln(1 + e^x) + C$$
Evaluate the indefinite integral $$\int$$cos(x)e^(sin(x)) dx.
e^(sin(x))/cos(x) + C
sin(x)·e^(sin(x)) + C
-e^(sin(x)) + C
e^(sin(x)) + C
$$ \displaystyle\int \frac{(5x^{1/6} - 1)^4}{x^{5/6}} dx = $$
$$\displaystyle\frac{4(5x^{1/6} - 1)^3}{x^{5/6}} + C$$
$$\displaystyle\frac{6(5x^{1/6} - 1)^5}{25} + C$$
$$\displaystyle\frac{(5x^{1/6} - 1)^5}{5} + C$$
$$\displaystyle\frac{6(5x^{1/6} - 1)^5}{30} + C$$
If $u = 3x + 7$, which of the following is the expression $\frac{2u^{3/2}}{9} + C$ written in terms of $x$?
(2x^(3/2))/9+C
(2(3x+7)^(3/2))/9+C(3x+7)
(2(3x+7)^(3/2))/9
(2(3x +7)^(3/2))/9+C