7.3 Trigonometric Substitution

1. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-3-1.html

-sqrt(-x^2 + 1)/x - arcsin(x)

2. .

   http://matrix.skku.ac.kr/cal-lab/cal-7-3-2.html

1/16*sqrt(2)*arcsinh(2*x/abs(2*x + 4) - 4/abs(2*x + 4)) +1/16*sqrt(2)*arcsinh(2*x/abs(2*x - 4) + 4/abs(2*x - 4))

3. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-3-3.html

-1/9*sqrt(x^2 + 9)/x

4. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-3-4.html

x/sqrt(-x^2 + 1)

5. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-3-5.html

1/3*sqrt(3)*log(2*sqrt(3*x^2 - 2*x - 1)*sqrt(3)+6*x -2)

6. .

http://matrix.skku.ac.kr/cal-lab/cal-7-3-6.html

-sqrt(-x^2 + 9)/x - arcsin(1/3*x)

7. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-3-7.html

1/9*x/sqrt(x^2 + 4*x + 13) + 2/9/sqrt(x^2 + 4*x + 13)

8. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-3-8.html

x/sqrt(-x^2+4)-arcsin(1/2*x)

9. .

   http://matrix.skku.ac.kr/cal-lab/cal-7-3-9.html

-1/3*arcsinh(3/abs(x))

10. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-3-10.html

sqrt(-x^2 + 9) - 3*log(6*sqrt(-x^2 + 9)/abs(x) + 18/abs(x))

11. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-3-11.html

 .

      .

      -1/2*a^2*log(2*x + 2*sqrt(-a^2 + x^2)) + 1/2*sqrt(-a^2 + x^2)*x

12. .

          http://matrix.skku.ac.kr/cal-lab/cal-7-3-12.html

1/6*(2*x^2 - 2)^(3/2)/x^3

13. .

          http://matrix.skku.ac.kr/cal-lab/cal-7-3-13.html

-1/2/(x^2 - 1)

14. .

          http://matrix.skku.ac.kr/cal-lab/cal-7-3-14.html

arcsin(x)

 15. .

          http://matrix.skku.ac.kr/cal-lab/cal-7-3-15.html

log(2*x + 2*sqrt(x^2 + 3*x + 1) + 3)

 16. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-3-16.html

-3*sqrt(-x^2-x+5) +1/2*arcsin(-1/21*(2*x + 1)*sqrt(21))

17. .

          http://matrix.skku.ac.kr/cal-lab/cal-7-3-17.html

sqrt(x^2 - 1) + arcsin(1/abs(x))

18. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-3-18.html

1/2*sqrt(-e^(2*x) + 9)*e^x + 9/2*arcsin(1/3*e^x)

19. (a) Prove that

       .

    (b) Use to prove that

       .

     http://matrix.skku.ac.kr/cal-lab/cal-7-3-19.html

 (a) , .

                      .

      1/2*a^2*arcsinh(x/sqrt(a^2)) + 1/2*sqrt(a^2 + x^2)*x

      (b) Use to prove that

         .