Substitution and Exponential Limits

MATH 116 SI worksheet · Winter 2024 · Session 1

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I. Rearrange the integrand if needed and find a suitable u-substitution

  1. \[\displaystyle \int x(1-x)^{99}\,dx\]
  2. \[\displaystyle \int \cos^3 x\,dx\]
  3. \[\displaystyle \int \sin^2 x\cos^3 x\,dx\]
  4. \[\displaystyle \int \frac{\sec^2 x}{\tan x+1}\,dx\]
  5. \[\displaystyle \int x\sqrt{x-5}\,dx\]
  6. \[\displaystyle \int \frac{x^2}{\sqrt{x-1}}\,dx\]
  7. \[\displaystyle \int x\sin(x^2)\cos(x^2)\,dx\]
  8. \[\displaystyle \int \sin x\cos^2 x\,(1-\cos^3 x)^{9}\,dx\]
  9. \[\displaystyle \int \frac{x^5}{(x^3-3)^{3/2}}\,dx\]
  10. \[\displaystyle \int \frac{dx}{1+e^x}\]
  11. \[\displaystyle \int \frac{x^2-1}{(x^4+3x^2+1)\arctan\left(\frac{x^2+1}{x}\right)}\,dx\]

II. Compound interest

An account starts with $1.00 and pays 100% interest per year. If the interest is credited once, at the end of the year, the account is worth $2.00 at year-end.

  • a. Write the compound interest formula.
  • b. Suppose interest is credited twice: 50% after six months, then 50% again at the end of the year. How much is in the account?
  • c. Suppose it is credited \(n\) times. Write the formula.
  • d. Evaluate your formula from c as \(n \to \infty\).

III. Evaluate the limits

  1. \[\displaystyle \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{7x}\]
  2. \[\displaystyle \lim_{x \to \infty} \left(1 + \frac{7}{x}\right)^{x}\]
  3. \[\displaystyle \lim_{x \to 0} \left(1 + 7x\right)^{1/x}\]
  4. \[\displaystyle \lim_{x \to \infty} \left(1 + \frac{1}{x^2}\right)^x\]
  5. \[\displaystyle \lim_{x \to \infty} \left(\frac{x}{1+x}\right)^x\]
  6. \[\displaystyle \lim_{x \to \infty} \left(\frac{2x^2+3}{2x^2+5}\right)^{8x^2+3}\]
  7. \[\displaystyle \lim_{x \to 0} \left(\frac{1+\tan x}{1+ \sin x}\right)^{1/\sin x}\]
  8. \[\displaystyle \lim_{x \to 1} \left(1 + \sin \pi x\right)^{\cot \pi x}\]

IV. If you have time: extra problems

  • a. For \(\displaystyle \lim_{x \to \infty} f(x)^{g(x)}\) with \(f(x) \to 1\) and \(g(x) \to \infty\), prove that the limit equals \(\displaystyle e^{\lim_{x \to \infty} g(x)\,[f(x) - 1]}.\)
  • b. Does \(\displaystyle \lim_{x \to \pi/2}(\sin x)^{1/\cos x}\) exist?
  • c. Find the antiderivatives of \(f:[0,2]\to \mathbb{R}\), \(\displaystyle f(x) = \sqrt{x^3+2-2\sqrt{x^3+1}}+\sqrt{x^3+10-6\sqrt{x^3+1}}.\)