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Do Now - #4 on p.328
Evaluate: 1tan ydy1tanu y dv dy
2
1
1du dy
y
v y
1 12
tan tan1
yydy y y dy
y
Integration by parts:
Now, use substitution to evaluate the new integral
Do Now - #4 on p.328
Evaluate: 1tan ydy1 1
2tan tan
1
yydy y y dy
y
21w y
2dw ydy1
2dw ydy1 1 1
tan2
y y dww
1 1
tan ln2
y y w C
1 21tan ln 1
2y y y C
Solving for the Unknown Integral
Section 6.3b
Practice Problems
Evaluate cosxe xdxxu exdu e dx
cosdv xdxsinv x
cos sin sinx x xe xdx e x e xdx xu exdu e dx
sindv xdxcosv x
sin cos cosx x xe x e x x e dx
Practice Problems
Evaluate cosxe xdx sin cos cosx x xe x e x x e dx
cos sin cos cosx x x xe xdx e x e x e xdx Now our unknown integral appears
on both sides of the equation!!!
2 cos sin cosx x xe xdx e x e x C Combine like terms:
Practice Problems
Evaluate cosxe xdx2 cos sin cosx x xe xdx e x e x C
Final Answer:sin cos
cos2
x xx e x e xe xdx C
Note: When using this technique, it is usually agood idea to keep the same choices for u and dvduring each step of the problem…
Practice Problems
Solve the differential equation:2 ln
dyx x
dx
lnu x 1du dx
x 2dv x dx 31
3v x
2 lndydx x x dx
dx
2 lny x x dxUse I.B.P. to evaluate this integral:
Practice Problems
Solve the differential equation:2 ln
dyx x
dx
2 lny x x dx 3 31 1 1ln
3 3x x x dx
x
3 21 1ln
3 3x x x dx 3 31 1ln
3 9x x x C
Practice Problems
Evaluate2 2
3sin 2xe xdx
2xu e
22 xdu e dx
sin 2dv xdx1cos 22
v x
2 21 1cos 2 cos 2 22 2
x xe x x e dx 2 21cos 2 cos 2
2x xe x e xdx
Practice Problems
Evaluate2 2
3sin 2xe xdx
2xu e22 xdu e dx
cos 2dv xdx1sin 22
v x
2 21cos 2 cos 2
2x xe x e xdx
21cos 2
2xe x
2 21 1sin 2 sin 2 2
2 2x xe x x e dx
Practice Problems
Evaluate2 2
3sin 2xe xdx
2 21cos 2 sin 2 sin 2
2x xe x x e xdx
2 212 sin 2 cos 2 sin 2
2x xe xdx e x x C
2 21sin 2 cos 2 sin 2
4x xe xdx e x x C Now, to apply the limits of integration…
Practice Problems
Evaluate2 2
3sin 2xe xdx
22
3
1cos2 sin 2
4xe x x
4 61 1cos 4 sin 4 cos 6 sin 6
4 4e e
125.028 Verify numerically!!!
