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
- Find \(\int 3x\cosh(x)\,dx\).
- Find \(\int te^{2t}\,dt\).
- Find \(\int y^2\sin(y)\,dy\).
- Find \(\int 4x\sec^2(x)\,dx\). (Hint: recall \(\int\tan\theta\,d\theta=\ln|\sec\theta|+C\).)
- Find \(\int e^{3w}\sinh(w)\,dw\).
- Find \(\int \sin(2x)\cos(4x)\,dx\).
- Compute \(\int_1^e x\ln x\,dx\).
- Find \(\int x^4e^x\,dx\).
- 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.)
- Find \(\int \cos^4 x\,dx\) using the above formula.
- Find \(\int x\cosh x\,dx\).
- Find \(\int e^\theta\sin\theta\,d\theta\).
Textbook References
- University Calculus: Early Transcendentals (3rd Ed)
- 8.1