\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

Section 2.1 Calculus 2

Integration by Substitution
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2.1 Integration by Substitution

2.1.1 Substitution and the Chain Rule

  • Reversing the Chain Rule \(\frac{d}{dx}[f(g(x))]=f’(g(x))g’(x)\) yields the Substitution Rule \(\int f’(g(x))g’(x)\,dx=f(g(x))+C\).
  • This is often abbreviated as \(\int f’(u)\,du=f(u)+C\) by using the substitutions \(u=g(x)\) and \(du=g’(x)dx\).
  • Example Find \(\int 4(3+4x)^2\,dx\).
  • Example Find \(\int 3u^2\sin(u^3)\,du\).
  • Example Find \(\int \frac{x}{x^2+1}\,dx\).
  • Example Find \(\int \frac{4\sinh(\ln t)}{t}\,dt\).

2.1.2 Substitution in Definite Integrals

  • When dealing with definite integrals, you may either convert the boundaries to \(u\)-values, or you must substitute back for the original variable before plugging in boundaries.
  • Example Compute \(\int_{1/4}^{1/2} 4(3+4x)^2\,dx\).
  • Example Compute \(\int_0^1 z\sqrt{1-z}\,dz\).
  • Example Compute \(\int_0^{\pi/4}\tan^2\theta\sec^2\theta\,d\theta\).

2.1.3 Antiderivatives of Trigonometric Functions

  • Example Use Substitution to find \(\int\tan\theta\,d\theta\).
  • The antiderivatives of the basic trig functions may be found by using Substitution.
    • \(\int\cos x\,dx = \sin x+C\).
    • \(\int\sin x\,dx = -\cos x+C\).
    • \(\int\sec x\,dx = \ln|\sec x+\tan x|+C\).
    • \(\int\csc x\,dx = -\ln|\csc x+\cot x|+C\).
    • \(\int\tan x\,dx = \ln|\sec x|+C\).
    • \(\int\cot x\,dx = -\ln|\csc x|+C\).
  • Example Prove that \(\int\csc x\,dx = -\ln|\csc x+\cot x|+C\).

Review Exercises

  1. Find \(\int 3(3x-5)^3\,dx\).
  2. Find \(\int 4e^{r-7}\,dr\).
  3. Find \(\int 4v\sech^2(2v^2+1)\,dv\).
  4. Find \(\int \frac{2e^x}{e^x+3}\,dx\).
  5. Find \(\int 2t^3\sqrt{t^2+1}\,dt\). (Hint: \(2t^3=2t\cdot t^2\).)
  6. Find \(\int \frac{2(\ln s)^3}{s}\,ds\).
  7. Find \(\int \frac{3\sqrt{x}}{2(x^{3/2}+2)^2}\,dx\).
  8. Find \(\int \frac{\cos(1/y)}{y^2}\,dy\).
  9. Compute \(\int_0^{\pi/12} \sec(3\theta)\tan(3\theta)\,d\theta\).
  10. Compute \(\int_1^2 (6x+3)(x^2+x)^2\,dx\).
  11. Compute \(\int_{\ln 3}^{\ln 8}e^z\sqrt{1+e^z}\,dz\).
  12. Compute \(\int_e^{e^2}\frac{1}{x\ln x}\,dx\).
  13. Use Substitution to find \(\int\cot\theta\,d\theta\).
  14. Multiply by \(\frac{\sec x+\tan x}{\sec x+\tan x}\) and use Substitution to prove \(\int\sec x\,dx=\ln|\sec x+\tan x|+C\).
  15. Find \(\int 3t^5(t^3+3)^2\,dt\).
  16. Evaluate \(\int_0^1 x^2e^{2x^3}\,dx\).

Solutions 1-8

Solutions 9-16

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 5.5, 5.6