Calculus Volume 1 · 5.6 Integrals Involving Exponential and Logarithmic Functions

Integrals Involving Exponential and Logarithmic Functions: practice problems

38 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

1. Exercises 321

   In the following exercises, compute each indefinite integral.

   $\int e^{-3x}dx$

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2. Exercises 323

   In the following exercises, compute each indefinite integral.

   $\int 3^{\text{-}x}dx$

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3. Exercises 325

   In the following exercises, compute each indefinite integral.

   $\int \frac{2}{x}dx$

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4. Exercises 327

   In the following exercises, compute each indefinite integral.

   $\int \frac{1}{\sqrt{x}}dx$

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5. Exercises 329

   In the following exercises, find each indefinite integral by using appropriate substitutions.

   $\int \frac{dx}{x{(\ln x)}^2}$

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6. Exercises 331

   In the following exercises, find each indefinite integral by using appropriate substitutions.

   $\int \frac{dx}{x\ln x\ln (\ln x)}$

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7. Exercises 333

   In the following exercises, find each indefinite integral by using appropriate substitutions.

   $\int \frac{\cos x-x\sin x}{x\cos x}dx$

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8. Exercises 335

   In the following exercises, find each indefinite integral by using appropriate substitutions.

   $\int \ln (\cos x)\tan xdx$

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9. Exercises 337

   In the following exercises, find each indefinite integral by using appropriate substitutions.

   $\int x^2{e}^{\text{-}{x}^3}dx$

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10. Exercises 339

    In the following exercises, find each indefinite integral by using appropriate substitutions.

    $\int e^{\tan x}{\sec }^{2}xdx$

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11. Exercises 341

    In the following exercises, find each indefinite integral by using appropriate substitutions.

    $\int \frac{e^{\ln (1-t)}}{1-t}dt$

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12. Exercises 342

    In the following exercises, verify by differentiation that $\int \ln xdx=x(\ln x-1)+C,$ then use appropriate changes of variables to compute the integral.

    $\int x\ln xdx$ $\text{(}Hint\text{:}\int x\ln xdx=\frac{1}{2}\int x\ln (x^2)dx)$

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13. Exercises 343

    In the following exercises, verify by differentiation that $\int \ln xdx=x(\ln x-1)+C,$ then use appropriate changes of variables to compute the integral.

    $\int x^2\text{ln(}{x}^2)dx$

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14. Exercises 345

    In the following exercises, verify by differentiation that $\int \ln xdx=x(\ln x-1)+C,$ then use appropriate changes of variables to compute the integral.

    $\int \frac{\ln x}{\sqrt{x}}dx$

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15. Exercises 347

    In the following exercises, verify by differentiation that $\int \ln xdx=x(\ln x-1)+C,$ then use appropriate changes of variables to compute the integral.

    Write an integral to express the area under the graph of $y=e^t$ between $t=0$ and and evaluate the integral.

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16. Exercises 349

    In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.

    $\int \frac{\sin (3x)-\cos (3x)}{\sin (3x)+\cos (3x)}dx$

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17. Exercises 351

    In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.

    $\int x\csc (x^2)dx$

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18. Exercises 353

    In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.

    $\int \ln (\csc x)\cot xdx$

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19. Exercises 355

    In the following exercises, evaluate the definite integral.

    ${\int }_1^2\frac{1+2x+x^2}{3x+3x^2+x^3}dx$

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20. Exercises 357

    In the following exercises, evaluate the definite integral.

    ${\int }_0^{\pi \text{/}3}\frac{\sin x-\cos x}{\sin x+\cos x}dx$

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21. Exercises 359

    In the following exercises, evaluate the definite integral.

    ${\int }_{\pi \text{/}4}^{\pi \text{/}3}\cot xdx$

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22. Exercises 361

    In the following exercises, integrate using the indicated substitution.

    $\int \frac{y-1}{y+1}dy;u=y+1$

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23. Exercises 363

    In the following exercises, integrate using the indicated substitution.

    $\int \frac{\sin x+\cos x}{\sin x-\cos x}dx;u=\sin x-\cos x$

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24. Exercises 365

    In the following exercises, integrate using the indicated substitution.

    $\int \ln (x)\frac{\sqrt{1-{(\ln x)}^2}}{x}dx;u=\ln x$

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25. Exercises 367

    In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R₅₀ and solve for the exact area.

    [T] $y=e^{\text{-}x}$ over $[0,1]$

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26. Exercises 369

    In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R₅₀ and solve for the exact area.

    [T] $y=\frac{x+1}{x^2+2x+6}$ over $[0,1]$

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27. Exercises 371

    In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R₅₀ and solve for the exact area.

    [T] $y=\text{-}{2}^{\text{-}x}$ over $[0,1]$

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28. Exercises 373

    In the following exercises, $f(x)\ge 0$ for $a\le x\le b.$ Find the area under the graph of $f(x)$ between the given values a and b by integrating.

    $f(x)=\frac{{\log }_2(x)}{x};a=32,b=64$

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29. Exercises 375

    In the following exercises, $f(x)\ge 0$ for $a\le x\le b.$ Find the area under the graph of $f(x)$ between the given values a and b by integrating.

    $f(x)=2^{\text{-}x};a=3,b=4$

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30. Exercises 377

    In the following exercises, $f(x)\ge 0$ for $a\le x\le b.$ Find the area under the graph of $f(x)$ between the given values a and b by integrating.

    Compute the integral of $f(x)=x{e}^{\text{-}{x}^2}$ and find the smallest value of such that the area under the graph between and is, at most, 0.01.

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31. Exercises 379

    In the following exercises, $f(x)\ge 0$ for $a\le x\le b.$ Find the area under the graph of $f(x)$ between the given values a and b by integrating.

    Show that ${\int }_a^b\frac{dt}{t}={\int }_{1\text{/}b}^{1\text{/}a}\frac{dt}{t}$ when

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32. Exercises 381

    In the following exercises, $f(x)\ge 0$ for $a\le x\le b.$ Find the area under the graph of $f(x)$ between the given values a and b by integrating.

    Use the previous exercise to find the antiderivative of $h(x)=x^x(1+\ln x)$ and evaluate

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33. Exercises 383

    The following exercises are intended to derive the fundamental properties of the natural log starting from the definition $\ln (x)={\int }_1^x\frac{dt}{t},$ using properties of the definite integral and making no further assumptions.

    Use the identity $\ln (x)={\int }_1^x\frac{dt}{t}$ to derive the identity

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34. Exercises 385

    The following exercises are intended to derive the fundamental properties of the natural log starting from the definition $\ln (x)={\int }_1^x\frac{dt}{t},$ using properties of the definite integral and making no further assumptions.

    Use the identity $\ln x={\int }_1^x\frac{dt}{t}$ to show that is an increasing function of on and use the previous exercises to show that the range of is Without any further assumptions, conclude that has an inverse function defined on

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35. Exercises 387

    The following exercises are intended to derive the fundamental properties of the natural log starting from the definition $\ln (x)={\int }_1^x\frac{dt}{t},$ using properties of the definite integral and making no further assumptions.

    Pretend, for the moment, that we do not know that $e^x$ is the inverse function of but keep in mind that has an inverse function defined on Call it . Show that

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36. Exercises 389

    The following exercises are intended to derive the fundamental properties of the natural log starting from the definition $\ln (x)={\int }_1^x\frac{dt}{t},$ using properties of the definite integral and making no further assumptions.

    [T] The normal distribution in probability is given by $p(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\text{-}{(x-\mu )}^2\text{/}2{\sigma }^2},$ where is the standard deviation and is the average. The in probability, corresponds to Compute the right endpoint estimates of

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37. Review Exercises 455

    Find the derivative.

    $\frac{d}{dt}{\int }_0^t\frac{\sin x}{\sqrt{1+x^2}}dx$

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38. Review Exercises 457

    Find the derivative.

    $\frac{d}{dx}{\int }_1^{\ln (x)}(4t+e^t)dt$

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