Exercises - Integration by Parts

  1. Use integration by parts to find each of the following:

    1. $\displaystyle{\int e^{4x} \sin(2x)\,dx}$

    2. $\displaystyle{\int (4x-3)\ln(2x-3)\,dx}$

    3. $\displaystyle{\int_0^2 2x^3 e^{x^2}\,dx}$

    4. $\displaystyle{\int \frac{1}{x^3}\sin\left(\frac{1}{x}\right)\,dx}$

    5. $\displaystyle{\int 9x\tan^2(3x)\,dx}$

    6. $\displaystyle{\int\sin(\sqrt{x})\,dx}$

    7. $\displaystyle{\int x^3\ln^2(x)\,dx}$

    8. $\displaystyle{\int \frac{\ln(x)}{\sqrt{x}}\,dx}$

    9. $\displaystyle{\int x\cos(\ln(x))\,dx}$

    10. $\displaystyle{\int x\cdot \textrm{arcsin }(x)\,dx}$

    11. $\displaystyle{\int \sin(2x)\cos(3x)\,dx}$

    12. $\displaystyle{\int \csc^3(x)\,dx}$

    13. $\displaystyle{\int \cos \sqrt{x}\,dx}$

    14. $\displaystyle{\int x \ln(x^2+1)\,dx}$

    1. $\displaystyle{-\frac{e^{4x}}{10}\cos(2x) + \frac{e^{4x}}{5}\sin(2x) + C}$

    2. $\displaystyle{(2x^2-3x)\ln(2x-3)-x^2 + C}$

    3. $\displaystyle{3e^4 + 1}$

    4. $\displaystyle{\frac{1}{x}\cos\left(\frac{1}{x}\right)-\sin\left(\frac{1}{x}\right) + C}$

    5. $\displaystyle{3x\tan(3x) + \ln|\cos(3x)| - \frac{9}{2}x^2 + C}$

    6. $\displaystyle{-2\sqrt{x} \cos(\sqrt{x}) + 2\sin(\sqrt{x}) + C}$

    7. $\displaystyle{\frac{x^4}{4}\ln^2(x) - \frac{x^4}{8}\ln(x) + \frac{x^4}{32} + C}$

    8. $\displaystyle{2\sqrt{x} \ln(x) - 4\sqrt{x} + C}$

    9. $\displaystyle{\frac{2}{5}x^2\cos(\ln(x)) + \frac{1}{5}x^2\sin(\ln(x)) + C}$

    10. $\displaystyle{\frac{1}{2} x^2 \textrm{arcsin }(x) - \frac{1}{4} \textrm{arcsin }(x) + \frac{1}{4}x \sqrt{1-x^2} + C}$

    11. $\displaystyle{\frac{3}{5}\sin(2x)\sin(3x)+\frac{2}{5}\cos(2x)\cos(3x)+C}$

    12. $\displaystyle{-\frac{1}{2}\csc(x)\cot(x) + \frac{1}{2}\ln|\csc(x)-\cot(x)| + C}$

    13. $\displaystyle{2\sqrt{x}\sin(\sqrt{x}) + 2\cos(\sqrt{x}) + C}$

    14. $\displaystyle{\frac{1}{2} x^2 \ln(x^2+1) - \frac{1}{2}(x^2+1) + \frac{1}{2}\ln|x^2+1| + C}$