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

Section 2.5 Calculus 2


Integration by Parts

2.5 Integration by Parts

2.5.1 Parts and the Product Rule

  • We may reorder the Product Rule \(\frac{d}{dx}[f(x)g(x)]=g(x)f’(x)+f(x)g’(x)\) as follows: \(f(x)g’(x)=\frac{d}{dx}[f(x)g(x)]-g(x)f’(x)\).
  • Integrating both sides yields the rule of Integration by Parts: \(\int f(x)g’(x)\,dx=f(x)g(x)-\int g(x)f’(x)\,dx\).
  • This is often abbreviated as \(\int u\,dv=uv-\int v\,du\) by using the substitutions \(u=f(x)\) \(du=f’(x)dx\), \(v=g(x)\) \(dv=g’(x)dx\).
  • Example Find \(\int 2x\cos(x)\,dx\).
  • Example Find \(\int te^t\,dt\).
  • Occasionally you’ll need to use parts twice.
  • Example Find \(\int 3x^2\sinh(x)\,dx\).
  • Especially tricky problems may involve cycling back to the original integral.
  • Example Find \(\int e^w\sin(2w)\,dw\).

2.5.2 Integrating Definite Integrals by Parts

  • When using parts to evaluate definite integrals, do not forget to apply the bounds of integration to the entire integral.
  • Example Find \(\int_0^1 s^2e^s\,ds\).

2.5.3 Antiderivatives of Logarithms

  • Integrating logarithms is based on integration by parts.
  • Example Use Integration by Parts to find \(\int\ln x\,dx\).

Review Exercises

  1. Find \(\int 3x\cosh(x)\,dx\).
  2. Find \(\int te^{2t}\,dt\).
  3. Find \(\int y^2\sin(y)\,dy\).
  4. Find \(\int 4x\sec^2(x)\,dx\). (Hint: recall \(\int\tan\theta\,d\theta=\ln|\sec\theta|+C\).)
  5. Find \(\int e^{3w}\sinh(w)\,dw\).
  6. Find \(\int \sin(2x)\cos(4x)\,dx\).
  7. Compute \(\int_1^e x\ln x\,dx\).
  8. Find \(\int x^4e^x\,dx\).
  9. Prove \( \int \cos^{n+2} (x)\,dx = \frac{\cos^{n+1} (x)\sin (x)}{n+2}+\frac{n+1}{n+2}\int\cos^n (x)\,dx \). (Hint: take the derivative of both sides.)
  10. Find \(\int \cos^4 x\,dx\) using the above formula.
  11. Find \(\int x\cosh x\,dx\).
  12. Find \(\int e^\theta\sin\theta\,d\theta\).

Solutions 1-5

Solutions 6-12


Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 8.1