Notes #10- Integration by Parts NEW PDF

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Professor Bianca Santoro...

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Notes #10: Integration by Parts Integration by parts is the reverse of the product rule. Here is the derivation below: d uv  uv 'vu ' dx d uv   vu ' uv ' dx

Now integrate the entire equation and we get:

d

dx (uv) dx   vu ' dx  uv ' dx

vu ' dx (uv ') dx (uv ') dx uv  vu ' dx

uv 

or this is the rule for integration by parts;

LIATE: is a pneumonic device that helps you choose your u. L: Logs I:Inverse trig functions such are arc sine etc A:Algebra (any polynomial expression such as x2) T:Trig E:Exponential function such as 2x or ex 1.

x sin x dx

2. ln x dx

2 x 3. x e dx

4. x cos xdx

5.

e

x

sin xdx

6.

x

2

sin x dx

x 7. e cos x dx

8.

3 x

x e dx

1

9.

arctan x dx 0

10.

e

x

cos 2x dx

2 11. x csc x dx

Ans:

e x cos 2x  2e x sin 2 x c 5

ln x 12.  2 dx x

1 13. sin x dx

Review of Integration: 1.

x

x  2 dx

ex 2.  x dx e 1

2 3. 3 x 4  x dx

2

5 x 3 4. e dx

7. 

dx 1  4 x2

4x 5. 3 dx

6.

4 5 8. x 3x  5 dx

sec 3 t  t dt

9.

10.

12.

 3

13) cot x dx   6

11.

14)

15)

16)

17)

18) Review of Derivatives: (please differentiate the following:)

2.y=arctan (5x 2)

1.sin(xy)=x

3.y= e

3 x5

5. y=53x

4.y=ln

6. Y=exsin5x2

3

4x  7

7)

8)...