Calculus Volume 1 · 6.7 Integrals, Exponential Functions, and Logarithms

Integrals, Exponential Functions, and Logarithms: practice problems

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

1. Exercises 295

   For the following exercises, find the derivative $\frac{dy}{dx}.$

   $y=\ln (2x)$

   Show the book’s answerSign up free to have it checked

2. Exercises 297

   For the following exercises, find the derivative $\frac{dy}{dx}.$

   $y=\frac{1}{\ln x}$

   Show the book’s answerSign up free to have it checked

3. Exercises 299

   For the following exercises, find the indefinite integral.

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

   Show the book’s answerSign up free to have it checked

4. Exercises 301

   For the following exercises, find the derivative $dy\text{/}dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

   [T] $y=x\ln (x)$

   Show the book’s answerSign up free to have it checked

5. Exercises 303

   For the following exercises, find the derivative $dy\text{/}dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

   [T] $y=\ln (\sin x)$

   Show the book’s answerSign up free to have it checked

6. Exercises 305

   For the following exercises, find the derivative $dy\text{/}dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

   [T] $y=7\ln (4x)$

   Show the book’s answerSign up free to have it checked

7. Exercises 307

   For the following exercises, find the derivative $dy\text{/}dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

   [T] $y=\ln (\tan x)$

   Show the book’s answerSign up free to have it checked

8. Exercises 309

   For the following exercises, find the derivative $dy\text{/}dx.$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

   [T] $y=\ln ({\cos }^{2}x)$

   Show the book’s answerSign up free to have it checked

9. Exercises 311

   For the following exercises, find the definite or indefinite integral.

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

   Show the book’s answerSign up free to have it checked

10. Exercises 313

    For the following exercises, find the definite or indefinite integral.

    ${\int }_0^2\frac{x^{3}dx}{x^2+1}$

    Show the book’s answerSign up free to have it checked

11. Exercises 315

    For the following exercises, find the definite or indefinite integral.

    ${\int }_2^e\frac{dx}{{x(\ln x)}^2}$

    Show the book’s answerSign up free to have it checked

12. Exercises 317

    For the following exercises, find the definite or indefinite integral.

    ${\int }_0^{\pi \text{/}4}\tan xdx$

    Show the book’s answerSign up free to have it checked

13. Exercises 319

    For the following exercises, find the definite or indefinite integral.

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

    Show the book’s answerSign up free to have it checked

14. Exercises 321

    For the following exercises, compute $dy\text{/}dx$ by differentiating $\ln y.$

    $y=\sqrt{x^2+1}\sqrt{x^2-1}$

    Show the book’s answerSign up free to have it checked

15. Exercises 323

    For the following exercises, compute $dy\text{/}dx$ by differentiating $\ln y.$

    $y=x^{-1\text{/}x}$

    Show the book’s answerSign up free to have it checked

16. Exercises 325

    For the following exercises, compute $dy\text{/}dx$ by differentiating $\ln y.$

    $y=x^e$

    Show the book’s answerSign up free to have it checked

17. Exercises 327

    For the following exercises, compute $dy\text{/}dx$ by differentiating $\ln y.$

    $y=\sqrt{x}\sqrt[3]{x}\sqrt[6]{x}$

    Show the book’s answerSign up free to have it checked

18. Exercises 329

    For the following exercises, compute $dy\text{/}dx$ by differentiating $\ln y.$

    $y=e^{\text{-}\ln x}$

    Show the book’s answerSign up free to have it checked

19. Exercises 331

    For the following exercises, evaluate by any method.

    ${\int }_1^{e^{\pi }}\frac{dx}{x}+{\int }_{-2}^{-1}\frac{dx}{x}$

    Show the book’s answerSign up free to have it checked

20. Exercises 333

    For the following exercises, evaluate by any method.

    $\frac{d}{dx}{\int }_x^{x^2}\frac{dt}{t}$

    Show the book’s answerSign up free to have it checked

21. Exercises 335

    For the following exercises, use the function $\ln x.$ If you are unable to find intersection points analytically, use a calculator.

    Find the area of the region enclosed by $x=1$ and $y=5$ above $y=\ln x.$

    Show the book’s answerSign up free to have it checked

22. Exercises 337

    For the following exercises, use the function $\ln x.$ If you are unable to find intersection points analytically, use a calculator.

    Find the area between $\ln x$ and the x-axis from $x=1\text{to}x=2.$

    Show the book’s answerSign up free to have it checked

23. Exercises 339

    For the following exercises, use the function $\ln x.$ If you are unable to find intersection points analytically, use a calculator.

    [T] Find the surface area of the shape created when rotating the curve in the previous exercise from $x=1$ to $x=2$ around the x-axis.

    Show the book’s answerSign up free to have it checked

24. Exercises 341

    If you are unable to find intersection points analytically in the following exercises, use a calculator.

    [T] Find the arc length of $y=1\text{/}x$ from $x=1\text{to}x=4.$

    Show the book’s answerSign up free to have it checked