Calculus Volume 1 · 5.7 Integrals Resulting in Inverse Trigonometric Functions

Integrals Resulting in Inverse Trigonometric Functions: practice problems

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

1. Exercises 255

   2. If $f=g\circ h,$ when reversing the chain rule, $\frac{d}{dx}(g\circ h)(x)=g^{\prime }\text{(}h(x))h^{\prime }\text{(}x),$ should you take $u=g(x)$ or $u=h(x)?$

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2. Exercises 257

   In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify f such that the integral takes the form $\int f(u)du.$

   For $x>1:\int \frac{x^2}{\sqrt{x-1}}dx=\frac{2}{15}\sqrt{x-1}(3x^2+4x+8)+C;u=x-1$

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3. Exercises 259

   In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify f such that the integral takes the form $\int f(u)du.$

   $\int \frac{x}{\sqrt{4x^2+9}}dx=\frac{1}{4}\sqrt{4x^2+9}+C;u=4x^2+9$

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4. Exercises 261

   In the following exercises, find the antiderivative using the indicated substitution.

   $\int {(x+1)}^{4}dx;u=x+1$

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5. Exercises 263

   In the following exercises, find the antiderivative using the indicated substitution.

   $\int {(2x-3)}^{-7}dx;u=2x-3$

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6. Exercises 265

   In the following exercises, find the antiderivative using the indicated substitution.

   $\int \frac{x}{\sqrt{x^2+1}}dx;u=x^2+1$

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7. Exercises 267

   In the following exercises, find the antiderivative using the indicated substitution.

   $\int (x-1){(x^2-2x)}^{3}dx;u=x^2-2x$

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8. Exercises 269

   In the following exercises, find the antiderivative using the indicated substitution.

   $\int {\cos }^3\theta d\theta ;u=\sin \theta$ $\text{(}Hint\text{:}{\cos }^2\theta =1-{\sin }^2\theta \text{)}$

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9. Exercises 271

   In the following exercises, use a suitable change of variables to determine the indefinite integral.

   $\int x{(1-x)}^{99}dx$

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10. Exercises 273

    In the following exercises, use a suitable change of variables to determine the indefinite integral.

    $\int {(11x-7)}^{-3}dx$

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11. Exercises 275

    In the following exercises, use a suitable change of variables to determine the indefinite integral.

    $\int {\cos }^3\theta \sin \theta d\theta$

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12. Exercises 277

    In the following exercises, use a suitable change of variables to determine the indefinite integral.

    $\int {\cos }^2(\pi t)\sin (\pi t)dt$

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13. Exercises 279

    In the following exercises, use a suitable change of variables to determine the indefinite integral.

    $\int t\sin (t^2)\cos (t^2)dt$

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14. Exercises 281

    In the following exercises, use a suitable change of variables to determine the indefinite integral.

    $\int \frac{x^2}{{(x^3-3)}^2}dx$

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15. Exercises 283

    In the following exercises, use a suitable change of variables to determine the indefinite integral.

    $\int \frac{y^5}{{(1-y^3)}^{3\text{/}2}}dy$

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16. Exercises 285

    In the following exercises, use a suitable change of variables to determine the indefinite integral.

    ${\int (1-{\cos }^3\theta )}^{10}{\cos }^2\theta \sin \theta d\theta$

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17. Exercises 287

    In the following exercises, use a suitable change of variables to determine the indefinite integral.

    $\int ({\sin }^2\theta -2\sin \theta ){({\sin }^3\theta -3{\sin }^2\theta )}^3\cos \theta d\theta$

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18. Exercises 289

    In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer.

    [T] $y=x{(1-x^2)}^3$ over $[-1,2]$

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19. Exercises 291

    In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer.

    [T] $y=\frac{x}{{(x^2+1)}^2}$ over $[-1,1]$

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20. Exercises 293

    In the following exercises, use a change of variables to evaluate the definite integral.

    ${\int }_0^1\frac{x}{\sqrt{1+x^2}}dx$

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21. Exercises 295

    In the following exercises, use a change of variables to evaluate the definite integral.

    ${\int }_0^1\frac{t^2}{\sqrt{1+t^3}}dt$

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22. Exercises 297

    In the following exercises, use a change of variables to evaluate the definite integral.

    ${\int }_0^{\pi \text{/}4}\frac{\sin \theta }{{\cos }^4\theta }d\theta$

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23. Exercises 299

    In the following exercises, evaluate the indefinite integral $\int f(x)dx$ with constant $C=0$ using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of C that would need to be added to the antiderivative to make it equal to the definite integral $F(x)={\int }_a^{x}f(t)dt,$ with the left endpoint of the given interval.

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24. Exercises 301

    In the following exercises, evaluate the indefinite integral $\int f(x)dx$ with constant $C=0$ using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of C that would need to be added to the antiderivative to make it equal to the definite integral $F(x)={\int }_a^{x}f(t)dt,$ with the left endpoint of the given interval.

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25. Exercises 303

    In the following exercises, evaluate the indefinite integral $\int f(x)dx$ with constant $C=0$ using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of C that would need to be added to the antiderivative to make it equal to the definite integral $F(x)={\int }_a^{x}f(t)dt,$ with the left endpoint of the given interval.

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26. Exercises 305

    In the following exercises, evaluate the indefinite integral $\int f(x)dx$ with constant $C=0$ using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of C that would need to be added to the antiderivative to make it equal to the definite integral $F(x)={\int }_a^{x}f(t)dt,$ with the left endpoint of the given interval.

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27. Exercises 307

    In the following exercises, use a change of variables to show that each definite integral is equal to zero.

    ${\int }_0^{\sqrt{\pi }}t\cos (t^2)\sin (t^2)dt$

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28. Exercises 309

    In the following exercises, use a change of variables to show that each definite integral is equal to zero.

    ${\int }_0^1\frac{1-2t}{(1+{(t-\frac{1}{2})}^2)}dt$

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29. Exercises 311

    In the following exercises, use a change of variables to show that each definite integral is equal to zero.

    ${\int }_0^2(1-t)\cos (\pi t)dt$

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30. Exercises 313

    In the following exercises, use a change of variables to show that each definite integral is equal to zero.

    Show that the average value of $f(x)$ over an interval $[a,b]$ is the same as the average value of $f(cx)$ over the interval $[\frac{a}{c},\frac{b}{c}]$ for

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31. Exercises 315

    In the following exercises, use a change of variables to show that each definite integral is equal to zero.

    Find the area under the graph of $g(t)=\frac{t}{{(1-t^2)}^a}$ between $t=0$ and where and is fixed. Evaluate the limit as

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32. Exercises 317

    In the following exercises, use a change of variables to show that each definite integral is equal to zero.

    The area of the top half of an ellipse with a major axis that is the x-axis from $x=-a$ to $x=a$ and with a minor axis that is the y-axis from $y=\text{-}b$ to $y=b$ can be written as ${\int }_{\text{-}a}^{a}b\sqrt{1-\frac{x^2}{a^2}}dx.$ Use the substitution to express this area in terms of an integral of a trigonometric function. You do not need to compute the integral.

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33. Exercises 319

    In the following exercises, use a change of variables to show that each definite integral is equal to zero.

    [T] The following graph is of a function of the form $f(x)=a\cos (nt)+b\cos (mt).$ Estimate the coefficients a and b and the frequency parameters n and m. Use these estimates to approximate ${\int }_0^{\frac{\pi }{2}}f(t)dt.$

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34. Exercises 391

    In the following exercises, evaluate each integral in terms of an inverse trigonometric function.

    ${\int }_0^{\sqrt{3}\text{/}2}\frac{dx}{\sqrt{1-x^2}}$

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35. Exercises 393

    In the following exercises, evaluate each integral in terms of an inverse trigonometric function.

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

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36. Exercises 395

    In the following exercises, evaluate each integral in terms of an inverse trigonometric function.

    ${\int }_{\frac{2}{\sqrt{3}}}^{\sqrt{2}}\frac{dx}{|x|\sqrt{x^2-1}}$

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37. Exercises 397

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

    $\int \frac{dx}{\sqrt{9-x^2}}$

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38. Exercises 399

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

    $\int \frac{dx}{9+x^2}$

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39. Exercises 401

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

    $\int \frac{dx}{|x|\sqrt{x^2-9}}$

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40. Exercises 403

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

    Explain the relationship $\text{-}{\cos }^{-1}t+C=\int \frac{dt}{\sqrt{1-t^2}}={\sin }^{-1}t+C.$ Is it true, in general, that

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41. Exercises 405

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

    Explain what is wrong with the following integral: ${\int }_1^2\frac{dt}{\sqrt{1-t^2}}.$

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42. Exercises 407

    In the following exercises, solve for the antiderivative $\int f$ of f with $C=0,$ then use a calculator to graph f and the antiderivative over the given interval $[a,b].$ Identify a value of C such that adding C to the antiderivative recovers the definite integral $F(x)={\int }_a^{x}f(t)dt.$

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43. Exercises 409

    In the following exercises, solve for the antiderivative $\int f$ of f with $C=0,$ then use a calculator to graph f and the antiderivative over the given interval $[a,b].$ Identify a value of C such that adding C to the antiderivative recovers the definite integral $F(x)={\int }_a^{x}f(t)dt.$

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44. Exercises 411

    In the following exercises, compute the antiderivative using appropriate substitutions.

    $\int \frac{{\sin }^{-1}tdt}{\sqrt{1-t^2}}$

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45. Exercises 413

    In the following exercises, compute the antiderivative using appropriate substitutions.

    $\int \frac{{\tan }^{-1}(2t)}{1+4t^2}dt$

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46. Exercises 415

    In the following exercises, compute the antiderivative using appropriate substitutions.

    $\int \frac{{\sec }^{-1}(\frac{t}{2})}{|t|\sqrt{t^2-4}}dt$

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47. Exercises 417

    In the following exercises, use a calculator to graph the antiderivative $\int f$ with $C=0$ over the given interval $[a,b].$ Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral $F(x)={\int }_a^{x}f(t)dt.$

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48. Exercises 419

    In the following exercises, use a calculator to graph the antiderivative $\int f$ with $C=0$ over the given interval $[a,b].$ Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral $F(x)={\int }_a^{x}f(t)dt.$

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49. Exercises 421

    In the following exercises, use a calculator to graph the antiderivative $\int f$ with $C=0$ over the given interval $[a,b].$ Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral $F(x)={\int }_a^{x}f(t)dt.$

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50. Exercises 423

    In the following exercises, compute each integral using appropriate substitutions.

    $\int \frac{e^t}{\sqrt{1-e^{2t}}}dt$

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51. Exercises 425

    In the following exercises, compute each integral using appropriate substitutions.

    $\int \frac{dt}{t\sqrt{1-{\ln }^{2}t}}$

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52. Exercises 427

    In the following exercises, compute each integral using appropriate substitutions.

    $\int \frac{{\cos }^{-1}(2t)}{\sqrt{1-4t^2}}dt$

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53. Exercises 429

    In the following exercises, compute each definite integral.

    ${\int }_0^{1\text{/}2}\frac{\tan ({\sin }^{-1}t)}{\sqrt{1-t^2}}dt$

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54. Exercises 431

    In the following exercises, compute each definite integral.

    ${\int }_0^{1\text{/}2}\frac{\sin ({\tan }^{-1}t)}{1+t^2}dt$

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55. Exercises 432

    In the following exercises, compute each definite integral.

    ${\int }_0^{1\text{/}2}\frac{\cos ({\tan }^{-1}t)}{1+t^2}dt$

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56. Exercises 433

    In the following exercises, compute each definite integral.

    For $A>0,$ compute $I(A)={\int }_{\text{-}A}^A\frac{dt}{1+t^2}$ and evaluate the area under the graph of on

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57. Exercises 435

    In the following exercises, compute each definite integral.

    Use the substitution $u=\sqrt{2}\cot x$ and the identity $1+{\cot }^{2}x={\csc }^{2}x$ to evaluate ( Multiply the top and bottom of the integrand by

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58. Exercises 437

    In the following exercises, compute each definite integral.

    [T] Approximate the points at which the graphs of $f(x)=x^2-1$ and ${g(x)=(x^2+1)}^{\frac{1}{2}}$ intersect to four decimal places, and approximate the area between their graphs to three decimal places.

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59. Review Exercises 447

    Evaluate the following integrals.

    ${\int }_{-1}^1(x^3-2x^2+4x)dx$

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60. Review Exercises 449

    Evaluate the following integrals.

    ${\int }_{\pi \text{/}3}^{\pi \text{/}2}2\sec (2\theta )\tan (2\theta )d\theta$

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