1. $\displaystyle\int{x\sin 3x\,\mathrm{d}x}$ 2. $\displaystyle\int{x\ln x\,\mathrm{d}x}$ 3. \(\

Question

1. $\displaystyle\int{x\sin 3x\,\mathrm{d}x}$ 2. $\displaystyle\int{x\ln x\,\mathrm{d}x}$ 3. $\displaystyle\int{\ln x\,\mathrm{d}x}$ 4. $\displaystyle\int{\frac{\ln x}{x}\,\mathrm{d}x}$ 5. $\displaystyle\int{\sin (\ln x)\,\mathrm{d}x}$ 6. $\displaystyle\int{{{2}^{\ln x}}\,\mathrm{d}x}$ 7. $\displaystyle\int{{{\tan }^{-1}}x\,\mathrm{d}x}$ 8. $\displaystyle\int{x{{\sin }^{2}}x\,\mathrm{d}x}$ 9. $\displaystyle\int{\frac{x}{{{\cos }^{2}}5x}\,\mathrm{d}x}$ 10. $\displaystyle\int{\text{cose}{{\text{c}}^{3}}x\,\mathrm{d}x}$ 11. $\displaystyle\int{\frac{1}{{{\cos }^{3}}3x}\,\mathrm{d}x}$ 12. $\displaystyle\int{\sec x\tan x{{e}^{\text{sec}x}}\,\mathrm{d}x}$

Accepted Answer

1. $-\frac{1}{3}x\cos 3x+\frac{1}{9}\sin 3x+C$ 2. $\frac{1}{2}{{x}^{2}}\ln x-\frac{1}{4}{{x}^{2}}+C$ 3. $x\ln x-x+C$ 4. $\frac{{{\left( \ln x ight)}^{2}}}{2}+C$ 5. $\frac{x}{2}\left[ \sin (\ln x)-\cos (\ln x) ight]+C$ 6. $\frac{x{{2}^{\ln x}}}{1+\ln 2}+C$ 7. $x{{	an }^{-1}}x-\frac{1}{2}\ln \left| 1+{{x}^{2}} ight|+C$ 8. $\frac{{{x}^{2}}}{4}-\frac{x\sin 2x}{4}-\frac{\cos 2x}{8}+C$ 9. $\frac{x	an 5x}{5}-\frac{\ln \left| \sec 5x ight|}{25}+C$ 10. $-\frac{1}{2}\left[ 	ext{cosec }x\cot x+\ln \left| 	ext{cosec }x+\cot x ight| ight]+C$ 11. $\frac{\sec 3x	an 3x}{6}+\frac{\ln \left| \sec 3x+	an 3x ight|}{6}+C$ 12. ${{e}^{	ext{sec }x}}+C$

Techniques of Integration - Infinite Integrals by Parts

Techniques of Integration - Infinite Integrals by Parts