7.1 Integration by Parts

1.

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

 ,

                 .

(log(x)^2 - 2*log(x) + 2)*x

2.

http://matrix.skku.ac.kr/cal-lab/cal-7-1-2.html

,

 ,

,

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2/49*(7*x-1)*e^(7*x) + 1/343*(49*x^2 - 14*x + 2)*e^(7*x) +1/7*e^(7*x)2/49*(7*x - 1)*e^(7*x) + 1/343*(49*x^2 - 14*x + 2)*e^(7*x) + 1/7*e^(7*x)

3.

 .

.

4.

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5.

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6.

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7.

8.

9.

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10.

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11.

.

12.

.

13.

.

14.

Thus,

.

Thus,

.

Thus,

.

Therefore,

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15.

16. 

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17.

18.

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Therefore,

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19.

http://matrix.skku.ac.kr/cal-lab/cal-7-1-19.html

.

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

20.

http://matrix.skku.ac.kr/cal-lab/cal-7-1-20.html

.

-1/10*(2*sin(2*x) - cos(2*x) + 5)*e^(-x)

21.

http://matrix.skku.ac.kr/cal-lab/cal-7-1-21.html

.

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

22.

http://matrix.skku.ac.kr/cal-lab/cal-7-1-22.html

.

1

23.

http://matrix.skku.ac.kr/cal-lab/cal-7-1-23.html

.

1/4*pi - 1/2*log(2)

 24.

http://matrix.skku.ac.kr/cal-lab/cal-7-1-24.html

.

2*pi - 4

25.

http://matrix.skku.ac.kr/cal-lab/cal-7-1-25.html

.

4*log(2)^2 - 2*log(2) + 15/32

26.

http://matrix.skku.ac.kr/cal-lab/cal-7-1-26.html

1/2*a^2*arcsin(x/sqrt(a^2)) + 1/2*sqrt(a^2 - x^2) * x1/2 * a^2 * arcsin(x/sqrt(a^2)) + 1/2*sqrt(a^2 - x^2)*x

27. (a) Establish the reduction formula. 

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(b) Use part (a) to evaluate .

(c) Use parts (a) and (b) to evaluate .

(a)

     .

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Therefore,

(b)

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(c)

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28. (a) Establish the reduction formula.

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(b) Find the integral explicitly for .

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29. (a) Establish the reduction formula

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(b) Find the integral for .

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