FAC1004: Rapid-Fire Drill Pack — Hyperbolic Differentiation & Integration

Objective: Achieve mechanical fluency in differentiating and integrating hyperbolic functions — pure pattern recognition and algebraic speed.
Target: 60–90 seconds per problem. If you stall >3 minutes, skip and mark it.
Total problems: 62
Estimated time: 1.5–2 hours (or two 45-minute sprints)

Cheat Sheet (Memorize First)

Exponential Definitions

$$\sinh x = \frac{e^x - e^{-x}}{2}, \quad \cosh x = \frac{e^x + e^{-x}}{2}, \quad \tanh x = \frac{\sinh x}{\cosh x}$$ $$\coth x = \frac{\cosh x}{\sinh x}, \quad \operatorname{sech} x = \frac{1}{\cosh x}, \quad \operatorname{csch} x = \frac{1}{\sinh x}$$

Fundamental Identity

$$\cosh^2 x - \sinh^2 x = 1, \quad 1 - \tanh^2 x = \operatorname{sech}^2 x, \quad \coth^2 x - 1 = \operatorname{csch}^2 x$$

Key Relations

$$\cosh x + \sinh x = e^x, \quad \cosh x - \sinh x = e^{-x}$$ $$\sinh(2x) = 2\sinh x \cosh x, \quad \cosh(2x) = \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 2\sinh^2 x + 1$$

Derivatives

Function Derivative
$\sinh u$ $\cosh u \cdot u'$
$\cosh u$ $\sinh u \cdot u'$
$\tanh u$ $\operatorname{sech}^2 u \cdot u'$
$\coth u$ $-\operatorname{csch}^2 u \cdot u'$
$\operatorname{sech} u$ $-\operatorname{sech} u \tanh u \cdot u'$
$\operatorname{csch} u$ $-\operatorname{csch} u \coth u \cdot u'$

Integrals

Integral Result
$\int \sinh u , du$ $\cosh u + C$
$\int \cosh u , du$ $\sinh u + C$
$\int \operatorname{sech}^2 u , du$ $\tanh u + C$
$\int \operatorname{csch}^2 u , du$ $-\coth u + C$
$\int \operatorname{sech} u \tanh u , du$ $-\operatorname{sech} u + C$
$\int \operatorname{csch} u \coth u , du$ $-\operatorname{csch} u + C$
$\int \tanh u , du$ $\ln(\cosh u) + C$
$\int \coth u , du$ $\ln\vert\sinh u\vert + C$

Inverse Hyperbolic Integration Forms

Integral Result
$\displaystyle\int \frac{du}{\sqrt{a^2 + u^2}}$ $\sinh^{-1}!\left(\frac{u}{a}\right) + C$
$\displaystyle\int \frac{du}{\sqrt{u^2 - a^2}}$ $\cosh^{-1}!\left(\frac{u}{a}\right) + C$
$\displaystyle\int \frac{du}{a^2 - u^2}$ $\frac{1}{a}\tanh^{-1}!\left(\frac{u}{a}\right) + C;;(\vert u\vert < a)\quad$ or $\quad\frac{1}{a}\coth^{-1}!\left(\frac{u}{a}\right) + C;;(\vert u\vert > a)$
$\displaystyle\int \frac{du}{u\sqrt{a^2 + u^2}}$ $-\frac{1}{a}\operatorname{csch}^{-1}!\left(\frac{u}{a}\right) + C$
$\displaystyle\int \frac{du}{u\sqrt{a^2 - u^2}}$ $-\frac{1}{a}\operatorname{sech}^{-1}!\left(\frac{u}{a}\right) + C$

Part A: Definitions, Identities & Basic Evaluation

Target: 60 seconds each. Pure recall and algebraic manipulation.

Set A1 — Evaluate & Simplify (8 problems)

  1. Express $\sinh(\ln 2)$ as a simplified fraction.
  2. Express $\cosh(\ln 3)$ as a simplified fraction.
  3. Simplify $\cosh^2 x - \sinh^2 x$ and state the identity.
  4. If $\sinh x = \frac{3}{4}$, find $\cosh x$ and $\tanh x$.
  5. If $\cosh x = \frac{5}{3}$ and $x > 0$, find $\sinh x$ and $\operatorname{sech} x$.
  6. Show that $\cosh x + \sinh x = e^x$ using exponential definitions.
  7. Simplify $e^{\sinh^{-1} x}$ in terms of $x$. (Hint: use the logarithmic form.)
  8. Show that $\sinh(2x) = 2\sinh x \cosh x$ using exponential definitions.

Score: ___/8

Set A2 — Identity Manipulation (6 problems)

  1. Prove $1 - \tanh^2 x = \operatorname{sech}^2 x$ starting from $\cosh^2 x - \sinh^2 x = 1$.
  2. If $\sinh x = t$, express $\cosh(2x)$ in terms of $t$.
  3. Simplify $\frac{\sinh x}{1 + \cosh x}$ into a single hyperbolic function.
  4. Solve for $x$: $2\cosh^2 x - 5\sinh x = 1$. (Hint: reduce to quadratic in $\sinh x$.)
  5. Show that $\ln(\cosh x + \sinh x) = x$.
  6. Prove $\coth^2 x - 1 = \operatorname{csch}^2 x$.

Score: ___/6

Part B: Derivatives of Hyperbolic Functions

Target: 60–90 seconds each. Apply derivative formula + chain rule.

Set B1 — Chain Rule Basics (8 problems)

Find $\frac{dy}{dx}$.

  1. $y = \sinh(3x + 2)$
  2. $y = \cosh(x^2)$
  3. $y = \tanh(5x)$
  4. $y = \coth(\sqrt{x})$
  5. $y = \operatorname{sech}(e^x)$
  6. $y = \operatorname{csch}(\ln x)$
  7. $y = \sinh(4x^3 - 2x)$
  8. $y = \cosh!\left(\frac{1}{x}\right)$

Score: ___/8

Set B2 — Composed & Nested Functions (8 problems)

Find $\frac{dy}{dx}$.

  1. $y = \ln(\cosh x)$
  2. $y = \sinh^3(2x)$
  3. $y = e^{\tanh x}$
  4. $y = \cosh(\sin x)$
  5. $y = \sinh(\ln x)$
  6. $y = \tanh^{-1}(\sinh x)$ (careful: chain rule on inverse tanh)
  7. $y = \sqrt{\cosh(2x)}$
  8. $y = \ln(\sinh(3x) + \cosh(3x))$ (simplify first using identity)

Score: ___/8

Set B3 — Product, Quotient & Implicit (8 problems)

Find $\frac{dy}{dx}$.

  1. $y = x^2 \sinh x$
  2. $y = \frac{\sinh x}{1 + \cosh x}$
  3. $y = e^x \tanh(2x)$
  4. $y = \frac{\cosh x}{x}$
  5. $y = \sinh x \cdot \cosh x$ (try two methods: product rule vs identity)
  6. $y = \ln!\left(\frac{\sinh x}{\cosh x}\right)$ (simplify first)
  7. $y = x \cosh x - \sinh x$
  8. $y = \coth(\ln x) \cdot \cos^{-1}(x^2)$

Score: ___/8

Part C: Integrals of Hyperbolic Functions

Target: 60–90 seconds each. Apply integral formula or $u$-substitution.

Set C1 — Direct Integrals (8 problems)

Evaluate.

  1. $\int \sinh(4x) , dx$
  2. $\int \cosh(2x - 1) , dx$
  3. $\int \operatorname{sech}^2(3x) , dx$
  4. $\int \operatorname{csch}^2!\left(\frac{x}{2}\right) , dx$
  5. $\int \tanh x , dx$ (Hint: write as $\frac{\sinh x}{\cosh x}$)
  6. $\int \coth x , dx$
  7. $\int \sinh x \cosh x , dx$ (try two methods)
  8. $\int \operatorname{sech} x \tanh x , dx$

Score: ___/8

Set C2 — $u$-Substitution (8 problems)

Evaluate.

  1. $\int \sinh^5 x \cosh x , dx$
  2. $\int \sqrt{\tanh x} \operatorname{sech}^2 x , dx$
  3. $\int x \sinh(x^2) , dx$
  4. $\int \coth^2 x \operatorname{csch}^2 x , dx$ (careful with sign from $du$)
  5. $\int \frac{\cosh x}{\sinh x} , dx$
  6. $\int \frac{\operatorname{sech}^2 x}{\tanh x} , dx$
  7. $\int \frac{\sinh x}{\sqrt{\cosh x}} , dx$
  8. $\int \cosh^3 x \sinh x , dx$

Score: ___/8

Set C3 — Definite Integrals (6 problems)

Evaluate.

  1. $\int_0^{\ln 2} \sinh x , dx$
  2. $\int_0^1 \cosh(2x) , dx$
  3. $\int_0^1 \frac{e^x - e^{-x}}{2} , dx$ (recognize the definition)
  4. $\int_0^1 \frac{dx}{\sqrt{1 + x^2}}$ (leads to $\sinh^{-1}$)
  5. $\int_0^{\ln 3} \tanh x , dx$
  6. $\int_{-1}^1 \sinh x , dx$ (use parity)

Score: ___/6

Part D: Integrals Leading to Inverse Hyperbolic Forms

Target: 90 seconds each. Recognize the standard form, perform $u$-substitution.

Set D1 — Standard Inverse Hyperbolic Integrals (8 problems)

Evaluate.

  1. $\int \frac{dx}{\sqrt{4 + x^2}}$
  2. $\int \frac{dx}{\sqrt{x^2 - 9}}$
  3. $\int \frac{dx}{25 - x^2}$ (specify domain: $\vert x\vert < 5$)
  4. $\int \frac{dx}{\sqrt{1 + 9x^2}}$
  5. $\int \frac{dx}{\sqrt{9x^2 - 16}}$
  6. $\int \frac{dx}{3 - 5x^2}$
  7. $\int \frac{dx}{x\sqrt{1 + x^2}}$
  8. $\int \frac{dx}{\sqrt{x^2 + 4x + 13}}$ (complete the square first)

Score: ___/8

Set D2 — Substitution Leading to Inverse Forms (6 problems)

Evaluate.

  1. $\int \frac{e^x}{\sqrt{e^{2x} + 1}} , dx$ (let $u = e^x$)
  2. $\int_0^{1/2} \frac{dx}{1 - 4x^2}$
  3. $\int \frac{dx}{\sqrt{(x+1)^2 + 4}}$
  4. $\int \frac{x}{\sqrt{x^4 - 1}} , dx$ (let $u = x^2$)
  5. $\int \frac{dx}{\sqrt{4x^2 + 12x + 5}}$ (factor inside root)
  6. $\int \frac{dx}{\sqrt{1 + e^{2x}}}$ (let $u = e^{-x}$)

Score: ___/6

Final Scorecard

Part Sets Problems Raw Score
A — Definitions & Identities A1, A2 14 ___/14
B — Hyperbolic Derivatives B1, B2, B3 24 ___/24
C — Hyperbolic Integrals C1, C2, C3 22 ___/22
D — Inverse Hyperbolic Integrals D1, D2 14 ___/14
TOTAL 74 ___/74

Proficiency Benchmarks

  • 55/74 (74%) — Proficient. You can handle standard exam problems.
  • 63/74 (85%) — Solid. Fast and accurate.
  • 69/74 (93%) — Exam-ready. Any mistake is a careless slip.

Speed Benchmarks

  • <1.5 hours: Excellent mechanical fluency. You own hyperbolic calculus.
  • 1.5–2 hours: Good. Review missed patterns.
  • >2 hours: Drill the specific sets you scored lowest on again tomorrow.

Error Log Template

After grading, list every wrong problem number with a one-word reason:

Problem Reason
e.g. 17 sign error
e.g. 47 forgot du factor

Re-solve all wrong problems immediately with notes, then again in 24 hours without notes.

Answer Key

  1. $\sinh(\ln 2) = \frac{2 - 1/2}{2} = \frac{3}{4}$
  2. $\cosh(\ln 3) = \frac{3 + 1/3}{2} = \frac{5}{3}$
  3. $\cosh^2 x - \sinh^2 x = 1$
  4. $\cosh x = \frac{5}{4}$, $\tanh x = \frac{3}{5}$
  5. $\sinh x = \frac{4}{3}$, $\operatorname{sech} x = \frac{3}{5}$
  6. $\frac{e^x - e^{-x}}{2} + \frac{e^x + e^{-x}}{2} = e^x$
  7. $e^{\sinh^{-1} x} = x + \sqrt{1 + x^2}$
  8. $\frac{e^{2x} - e^{-2x}}{2} = 2 \cdot \frac{e^x - e^{-x}}{2} \cdot \frac{e^x + e^{-x}}{2} = 2\sinh x \cosh x$
  9. Divide $\cosh^2 x - \sinh^2 x = 1$ by $\cosh^2 x$
  10. $\cosh(2x) = 1 + 2\sinh^2 x = 1 + 2t^2$
  11. $\tanh(x/2)$
  12. $\sinh x = \frac{1}{2}$ or $-3$ (reject); $x = \sinh^{-1}(1/2)$
  13. $\ln(e^x) = x$
  14. Divide $\cosh^2 x - \sinh^2 x = 1$ by $\sinh^2 x$
  15. $3\cosh(3x+2)$
  16. $2x \sinh(x^2)$
  17. $5\operatorname{sech}^2(5x)$
  18. $-\frac{1}{2\sqrt{x}}\operatorname{csch}^2(\sqrt{x})$
  19. $-\operatorname{sech}(e^x) \tanh(e^x) \cdot e^x$
  20. $-\frac{1}{x}\operatorname{csch}(\ln x) \coth(\ln x)$
  21. $(12x^2 - 2)\cosh(4x^3 - 2x)$
  22. $-\frac{1}{x^2}\sinh(1/x)$
  23. $\tanh x$
  24. $3\sinh^2(2x) \cdot 2\cosh(2x) = 6\sinh^2(2x)\cosh(2x)$
  25. $e^{\tanh x} \operatorname{sech}^2 x$
  26. $-\sinh(\sin x) \cdot \cos x$
  27. $\frac{\cosh(\ln x)}{x} = \frac{x + 1/x}{2x} = \frac{x^2 + 1}{2x^2}$
  28. $\frac{\cosh x}{1 - \sinh^2 x} = \frac{\cosh x}{\cosh^2 x} = \operatorname{sech} x$
  29. $\frac{1}{2\sqrt{\cosh(2x)}} \cdot 2\sinh(2x) = \frac{\sinh(2x)}{\sqrt{\cosh(2x)}}$
  30. $\ln(e^{3x}) = 3x$, so derivative $= 3$
  31. $2x\sinh x + x^2\cosh x$
  32. $\frac{\cosh x(1 + \cosh x) - \sinh x(\sinh x)}{(1 + \cosh x)^2} = \frac{\cosh x + 1}{(1 + \cosh x)^2} = \frac{1}{1 + \cosh x}$
  33. $e^x\tanh(2x) + 2e^x \operatorname{sech}^2(2x)$
  34. $\frac{x\sinh x - \cosh x}{x^2}$
  35. $\sinh^2 x + \cosh^2 x = \cosh(2x)$, or $\frac{1}{2}\sinh(2x)$ derivative $= \cosh(2x)$
  36. $\ln(\tanh x)$, derivative $= \frac{\operatorname{sech}^2 x}{\tanh x} = \frac{1}{\sinh x \cosh x} = \operatorname{csch} x \operatorname{sech} x$
  37. $\cosh x + x\sinh x - \cosh x = x\sinh x$
  38. $-\frac{1}{x}\operatorname{csch}^2(\ln x) \cdot \cos^{-1}(x^2) - \frac{2x}{\sqrt{1-x^4}}\coth(\ln x)$
  39. $\frac{1}{4}\cosh(4x) + C$
  40. $\frac{1}{2}\sinh(2x-1) + C$
  41. $\frac{1}{3}\tanh(3x) + C$
  42. $-2\coth(x/2) + C$
  43. $\ln(\cosh x) + C$
  44. $\ln\vert\sinh x\vert + C$
  45. $\frac{1}{2}\sinh^2 x + C$ or $\frac{1}{2}\cosh^2 x + C$ (differ by constant)
  46. $-\operatorname{sech} x + C$
  47. $\frac{1}{6}\sinh^6 x + C$
  48. $\frac{2}{3}\tanh^{3/2} x + C$
  49. $\frac{1}{2}\cosh(x^2) + C$
  50. $-\frac{1}{3}\coth^3 x + C$
  51. $\ln\vert\sinh x\vert + C$
  52. $\ln\vert\tanh x\vert + C$
  53. $2\sqrt{\cosh x} + C$
  54. $\frac{1}{4}\cosh^4 x + C$
  55. $\cosh(\ln 2) - \cosh 0 = \frac{5}{3} - 1 = \frac{2}{3}$
  56. $\frac{1}{2}\sinh 2$
  57. $\int_0^1 \sinh x , dx = \cosh 1 - 1$
  58. $\sinh^{-1} 1 = \ln(1 + \sqrt{2})$
  59. $\ln(\cosh(\ln 3)) - \ln(\cosh 0) = \ln(5/3) - \ln 1 = \ln(5/3)$
  60. $0$ (odd function over symmetric interval)
  61. $\sinh^{-1}(x/2) + C$
  62. $\cosh^{-1}(x/3) + C$
  63. $\frac{1}{5}\tanh^{-1}(x/5) + C$
  64. $\frac{1}{3}\sinh^{-1}(3x) + C$
  65. $\frac{1}{3}\cosh^{-1}(3x/4) + C$
  66. $\frac{1}{\sqrt{5}}\tanh^{-1}!\left(\frac{\sqrt{5}x}{\sqrt{3}}\right) + C$ or $\frac{1}{2\sqrt{15}}\ln!\left\vert\frac{\sqrt{3} + \sqrt{5}x}{\sqrt{3} - \sqrt{5}x}\right\vert + C$
  67. $-\operatorname{csch}^{-1}(x) + C$ or $-\ln!\left\vert\frac{1 + \sqrt{1 + x^2}}{x}\right\vert + C$
  68. $\int \frac{dx}{\sqrt{(x+2)^2 + 9}} = \sinh^{-1}!\left(\frac{x+2}{3}\right) + C$
  69. $\sinh^{-1}(e^x) + C$
  70. $\frac{1}{2}\tanh^{-1}(2x)\big|_0^{1/2}$ diverges (approaches $\infty$ — improper)
  71. $\sinh^{-1}!\left(\frac{x+1}{2}\right) + C$
  72. $\frac{1}{2}\cosh^{-1}(x^2) + C$
  73. $\frac{1}{2}\cosh^{-1}!\left(\frac{x+3/2}{\sqrt{5}/2}\right) + C$
  74. $-\operatorname{csch}^{-1}(e^{x}) + C$ or $-\ln!\left(e^{-x} + \sqrt{1 + e^{-2x}}\right) + C$

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