4. Integration by Parts

Recall:       \(\displaystyle \int u\,dv=u\,v-\int v\,du\)   where   \(du=\dfrac{du}{dx}\,dx\)   and   \(dv=\dfrac{dv}{dx}\,dx\)
For definite integrals: \(\displaystyle \int_a^b u\,dv=\left[\rule{0pt}{10pt}u\,v\right]_a^b-\int_a^b v\,du\)

Homework

Use integration by parts to compute the following indefinite integrals. Be sure to check your answer by differentiating.

1. \(\displaystyle \int x\cos(2x)\,dx\)

2. \(\displaystyle \int \dfrac{3x+1}{e^{2x-1}}\,dx\)

3. \(\displaystyle \int x^2e^{4x}\,dx\)

4. \(\displaystyle \int (x^3+3x^2)\sin 3x\,dx\)

5. \(\displaystyle \int x(x-2)^3\,dx\)   Compute once by parts and once by substitution. Check they give the same answer.

6. \(\displaystyle \int y^3\cos(y^2)\,dy\)

7. \(\displaystyle \int (3p^2+2p)\ln p\,dp\)

8. \(\displaystyle \int x\,\text{arcsec}\,x\,dx\) for \(x \ge 1\)

9. \(\displaystyle \int \ln(x^3)\,dx\)

---

10. An important function in physics is the sine integral defined by \[ \mathrm{Si}(x)=\int_0^x \dfrac{\sin t}{t}\,dt \] There is no exact formula for the result of this integral; it must be computed numerically. However, its derivative is easily found by the fundamental theorem of calculus: \[ \dfrac{d}{dx}\mathrm{Si}(x) =\dfrac{\sin x}{x} \] In addition, its antiderivative is easily found by integration by parts.
    Compute \(\displaystyle \int \mathrm{Si}(x)\,dx\).

---

Use integration by parts to compute the following definite integrals. Be sure to check your indefinite integral by differentiating.

11. \(\displaystyle \int_0^2 2x^2e^{2x}\,dx\)

12. \(\displaystyle \int_1^3 x^2\ln x\,dx\)

---

13. Sketch the region bounded by the curves \(y=\ln x\), \(y=-\ln x\) and \(x=e\). Then find the area of the region.

14. Find the average value of the function \(f(x)=x e^{-x}\) over the interval \([0,2]\).

    

---

Use integration by parts to compute the following indefinite integrals. Be sure to check your answer by differentiating.

15. \(\displaystyle \int e^{3x}\cos(4x)\,dx\)

16. \(\displaystyle \int \sin(3x)\cos(4x)\,dx\)

---

17. Use the the sine reduction formula: \[ \int \sin^n x\,dx =-\,\dfrac{1}{n}\sin^{n-1}x\cos x+\dfrac{n-1}{n}\int \sin^{n-2}x\,dx \] Compute \(\displaystyle \int \sin^3 x\,dx\).

---

18. Find the area of the region below \(y=x\cos x\) above the \(x\)-axis between \(x=0\) to \(x=\dfrac{\pi}{2}\).