Exponential & Logarithmic Integrals

Integrating exponential, logarithmic, and reciprocal (1/x) functions.

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Exponential & Logarithmic Integrals

22 questions

AP Learning Target(s): LT 8: Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

Antiderivative of the Reciprocal Function

Which of the following is the general antiderivative of $$\frac{1}{x}$$?

  1. A.

    $$\frac{x^0}{0} + C$$

  2. B.

    $$x^{-1} + C$$

  3. C.

    $$\ln(x) + C$$

  4. D.

    $$\ln\left|x\right| + C$$

Correct answer: D
Antiderivative With Power and Log Rules

I. $$\int \left(7\sqrt{x} + \frac{4}{x}\right) dx = \frac{14}{3}x^{3/2} + 4\ln|x| + C$$

II. $$\int \left(7\sqrt{x} + \frac{4}{x}\right) dx = \frac{7}{3}x^{3/2} + 4\ln|x| + C$$

III. $$\int \left(7\sqrt{x} + \frac{4}{x}\right) dx = \frac{14}{3}x^{3/2} + 4x + C$$

Which of the statements above is true?

  1. A.

    III only

  2. B.

    I and III

  3. C.

    I only

  4. D.

    II only

Correct answer: C
Definite Integral of an Exponential Function

Evaluate $$\displaystyle\int_{0}^{1} 6^x dx$$.

  1. A.

    $$\displaystyle\frac{5}{\ln 6}$$

  2. B.

    $$5$$

  3. C.

    $$\displaystyle\frac{6}{\ln 6}$$

  4. D.

    $$\displaystyle\frac{5}{\ln 5}$$

Correct answer: A
Evaluating an Improper Logarithmic Integral

Evaluate the improper integral $$\int_{2}^{\infty} \frac{1}{x \ln x} dx$$

  1. A.

    Converges to 2

  2. B.

    Converges to 1

  3. C.

    Converges to 0

  4. D.

    Diverges

Correct answer: D
Exponential Function Integration

Evaluate $$\displaystyle\int \frac{1}{2} \cdot 10^{-3x+2} dx$$.

  1. A.

    $$\displaystyle\frac{10^{-3x+2}}{6 \ln 10} + C$$

  2. B.

    $$\displaystyle-\frac{3(\ln 10) \cdot 10^{-3x+2}}{2} + C$$

  3. C.

    $$\displaystyle\frac{1}{2} \cdot 10^{-3x+2} + C$$

  4. D.

    $$\displaystyle-\frac{10^{-3x+2}}{6 \ln 10} + C$$

Correct answer: D
Exponential Function Integration

Evaluate $$\displaystyle\int e^{4x} dx$$.

  1. A.

    $$\displaystyle\frac{e^{4x}}{4x} + C$$

  2. B.

    $$e^{4x} + C$$

  3. C.

    $$4e^{4x} + C$$

  4. D.

    $$\displaystyle\frac{1}{4}e^{4x} + C$$

Correct answer: D
Indefinite Integral of an Exponential Function

Evaluate $$\displaystyle \int 6e^{3x} dx$$.

  1. A.

    $$2e^{3x} + C$$

  2. B.

    $$18e^{3x} + C$$

  3. C.

    $$6e^{3x} + C$$

  4. D.

    $$\displaystyle \frac{6e^{3x+1}}{3x + 1} + C$$

Correct answer: A
Indefinite Integral of an Exponential Function

Evaluate $$\displaystyle\int 8^{x/2} dx$$

  1. A.

    $$8^{x/2} + C$$

  2. B.

    $$\displaystyle\frac{(\ln 8) \cdot 8^{x/2}}{2} + C$$

  3. C.

    $$\displaystyle\frac{8^{x/2}}{2 \ln 8} + C$$

  4. D.

    $$\displaystyle\frac{2 \cdot 8^{x/2}}{\ln 8} + C$$

Correct answer: D
Indefinite Integral of an Exponential Function

Evaluate $$\displaystyle\int 7^{-2x} dx$$

  1. A.

    $$-2(\ln 7) \cdot 7^{-2x} + C$$

  2. B.

    $$7^{-2x} + C$$

  3. C.

    $$\displaystyle\frac{7^{-2x}}{2 \ln 7} + C$$

  4. D.

    $$-\displaystyle\frac{7^{-2x}}{2 \ln 7} + C$$

Correct answer: D
Indefinite Integral of an Exponential Function

Evaluate $$\displaystyle\int 5 \cdot 3^x dx$$.

  1. A.

    $$\displaystyle\frac{15^x}{\ln 15} + C$$

  2. B.

    $$5(\ln 3) \cdot 3^x + C$$

  3. C.

    $$\displaystyle\frac{5 \cdot 3^x}{\ln 3} + C$$

  4. D.

    $$5 \cdot 3^x + C$$

Correct answer: C
Indefinite Integral With Exponential and Cosine

$$\int$$(5e^x - 2cos(x)) dx =

  1. A.

    5e^x + 2sin(x)

  2. B.

    5e^x + 2cos(x)

  3. C.

    5e^x - 2sin(x) + C

  4. D.

    5e^x - 2cos(x)

Correct answer: C
Integral of an Exponential Function

Evaluate $$\int 2^x dx$$.

  1. A.

    $$2^x + C$$

  2. B.

    $$\dfrac{2^x}{\ln 2} + C$$

  3. C.

    $$\dfrac{2^{x+1}}{x+1} + C$$

  4. D.

    $$(\ln 2) 2^x + C$$

Correct answer: B
Integral of an Exponential Function

Evaluate $${\displaystyle\int} 9^{2x} dx$$

  1. A.

    $$9^{2x} + C$$

  2. B.

    $$\displaystyle\frac{9^{2x}}{2 \ln 9} + C$$

  3. C.

    $$2(\ln 9) \cdot 9^{2x} + C$$

  4. D.

    $$\displaystyle\frac{9^{2x}}{\ln 9} + C$$

Correct answer: B
Integrating an Exponential Function

Evaluate $$\displaystyle\int e^{2x+3} dx$$.

  1. A.

    $$\displaystyle 2e^{2x+3} + C$$

  2. B.

    $$e^{2x+3} + C$$

  3. C.

    $$\displaystyle\frac{1}{2}e^{2x+3} + C$$

  4. D.

    $$2e^{2x+3} + C$$

Correct answer: C
Integrating an Exponential Function

Evaluate $$\displaystyle\int \frac{1}{3} \cdot 4^{3x-1} dx$$

  1. A.

    $$\displaystyle\frac{4^{3x-1}}{9 \ln 4} + C$$

  2. B.

    $$\displaystyle\frac{4^{3x-1}}{3 \ln 4} + C$$

  3. C.

    $$\displaystyle\frac{(\ln 4) \cdot 4^{3x-1}}{3} + C$$

  4. D.

    $$\displaystyle\frac{1}{3} \cdot 4^{3x-1} + C$$

Correct answer: A
Integrating an Exponential Function and a Constant

Evaluate the integral $$\displaystyle\int \left(2^t + e^\pi \right) dt$$.

  1. A.

    $$\displaystyle\dfrac{2^t}{\ln 2} + e^\pi + C$$

  2. B.

    $$\displaystyle\dfrac{2^t}{\ln 2} + e^\pi t + C$$

  3. C.

    $$\displaystyle 2^t \ln 2 + e^\pi t + C$$

  4. D.

    $$\displaystyle 2^t \ln 2 + \dfrac{e^{\pi+1}}{\pi + 1} + C$$

Correct answer: B
Integrating Functions With Natural Logarithms

Which of the following methods is a common technique for integrating functions that include a natural logarithm term?

  1. A.

    Power rule

  2. B.

    Partial fractions

  3. C.

    u-substitution

  4. D.

    Logarithmic differentiation

Correct answer: D
Integration of Exponential and Sine Functions

Evaluate $$\int$$(2e^x - 3sin(x)) dx.

  1. A.

    2e^x + 3sin(x)

  2. B.

    2e^x - 3cos(x)

  3. C.

    2e^x + 3cos(x) + C

  4. D.

    2e^x - 3sin(x)

Correct answer: C
Integration of Exponential Functions

Evaluate the integral $$\displaystyle\int (3^x - 2^x) dx$$.

  1. A.

    $$\displaystyle\frac{3^x - 2^x}{\ln 3 - \ln 2} + C$$

  2. B.

    $$\displaystyle\frac{3^x}{\ln 3} - \frac{2^x}{\ln 2} + C$$

  3. C.

    $$3^x - 2^x + C$$

  4. D.

    $$(\ln 3) \cdot 3^x - (\ln 2) \cdot 2^x + C$$

Correct answer: B
Integration Technique for a Logarithmic Function

Which integration technique is required to evaluate the integral $$\int (\ln(x))^2 dx$$?

  1. A.

    U-substitution with u = ln(x)

  2. B.

    Partial fraction decomposition

  3. C.

    Integration by parts applied twice

  4. D.

    Trigonometric substitution

Correct answer: C
Integration Technique for an Exponential Function

Which of the following integration techniques is the most appropriate for finding the antiderivative of $$e^{2x}$$?

  1. A.

    Substitution

  2. B.

    Integration by parts

  3. C.

    Trigonometric substitution

  4. D.

    Partial fraction decomposition

Correct answer: A
Logarithmic Integration Rule

Evaluate $$\displaystyle\int_{1}^{e} \frac{2x + 3}{x^2 + 3x} dx$$

  1. A.

    $$2e + 3 - \ln(e + 3)$$

  2. B.

    $$e^2 + 3e - 4$$

  3. C.

    $$\frac{e^2 + 3e}{4}$$

  4. D.

    $$\ln(e^2 + 3e) - \ln 4$$

Correct answer: D