# 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