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Integration by Parts Exercises


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  1. {$ \large \int xe^{x^2} dx $}
  2. {$ \large \int t\ \sin\ t\ dt $}
  3. {$ \large \int x^2e^{2x} dx $}
  4. {$ \large \int \arcsin\ x\ dx $}
  5. {$ \large \int x^3\log\ x \ dx$}
  6. {$ \large \int u\sqrt{1-u}\ dx$}
  7. {$ \large \int x^2\ \arcsin\ x\ dx $}
  8. {$ \large \int x\ \tan^2x\ dx $}
  9. {$ \large \int x^3\sqrt{a^2-x^2}\ dx$}
  10. {$ \large \int x^3\sqrt{a^4-x^4}\ dx$}
  11. {$ \large \int \cos \theta\ \log\sin\theta\ d\theta$}
  12. {$ \large \int \sec^4\theta\ d\theta$}
  13. {$ \large \int \cos^3\theta\ d\theta$}
  14. {$ \large \int \frac{x^2dx}{(a^2-x^2)^{3 \over 2} } $}
  15. {$ \large \int \frac{x^2dx}{(1-x^2)^2 } $}
  16. {$ \large \int \frac{y^3dy}{a^2-y^2 } $}

Selected Additional Exercises

  • {$ \large \int x\ \cos\ x\ dx $}
  • {$ \large \int x^2e^x dx $}
  • {$ \large \int x\ \log\ x \ dx$}
  • {$ \large \int \arctan\ x\ dx $}
  • {$ \large \int x^2 \arctan\ x\ dx $}
  • {$ \large \int \frac{x(e^x-e^{-x})}{2} dx $}
  • {$ \large \int x^3e^{3x} dx $}
  • {$ \large \int e^x\ \sin\ x\ dx $}
  • {$ \large \int e^{2x}\ \cos\ 3x\ dx $}
  • {$ \large \int e^{-x}\ \sin\ 3x\ dx $}
  • {$ \large \int e^{ax}\ \cos\ nx\ dx $}
  • {$ \large \int (2-3x+4x^2)\ \log\ x \ dx$}
  • {$ \large \int (3x^2+4x+1)\ \arctan\ x \ dx$}
  • {$ \large \int (x^2-x)\ e^{-2x} \ dx$}
  • {$ \large \int (x^3-5)\ e^{4x} \ dx$}
  • Show {$ \large \int \frac{du}{v} = \frac{u}{v} + \int \frac{u}{v^2}\ dv$}
  • {$ \large \int e^{ax}\ \sin\ nx\ dx $}
  • {$ \large \int_1^e \log\ x \ dx$}
  • {$ \large \int_{-1}^2 xe^{-x} \ dx$}
  • {$ \large \int_0^{\frac{1}{2}} \arcsin\ x \ dx$}
  • {$ \large \int_0^1 (1+3x^2) \arctan\ x \ dx$}
  • {$ \large \int_0^{\frac{\pi}{2}} e^{-2x}\cos\ 3x \ dx$}
  • {$ \large \int_1^2 (e^x-e^{-x}) \ dx$}

Sources:

  • Love, Clyde E., Earl David Rainville Differential and Integral Calculus (Macmillian, 1916) Google Books p. 157, folio 132.
  • Davis,Ellery Williams, William Charles Brenke, Earle Raymond Hedrick. The Calculus (Macmillan Company, 1922) Google Books p. 182, folio 163.

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