Do Now - #4 on p.328Evaluate: 1tan ydy

1tanu y dv dy

2

11

du dyy

v y

1 12tan tan

1yydy y y dyy

Integration by parts:

Now, use substitution to evaluate the new integral

Do Now - #4 on p.328Evaluate: 1tan ydy

1 12tan tan

1yydy y y dyy

21w y 2dw ydy

12dw ydy1 1 1tan

2y y dw

w 1 1tan ln

2y y w C

1 21tan ln 12

y y y C

Solving for the Unknown IntegralSection 6.3b

Practice ProblemsEvaluate cosxe xdx

xu 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 ProblemsEvaluate 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 appearson both sides of the equation!!!

2 cos sin cosx x xe xdx e x e x C Combine like terms:

Practice ProblemsEvaluate cosxe xdx2 cos sin cosx x xe xdx e x e x C

Final Answer:sin coscos

2

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 lndy x x

dx

lnu x 1du dxx

2dv x dx 313

v x

2 lndy dx x x dxdx

2 lny x x dx

Use I.B.P. to evaluate this integral:

Practice Problems

Solve the differential equation:2 lndy x x

dx

2 lny x x dx 3 31 1 1ln

3 3x x x dx

x

3 21 1ln3 3x x x dx 3 31 1ln

3 9x x x C

Practice ProblemsEvaluate

2 2

3sin 2xe xdx

2xu e

22 xdu e dx

sin 2dv xdx1 cos22

v x

2 21 1cos 2 cos2 22 2

x xe x x e dx 2 21 cos2 cos2

2x xe x e xdx

Practice ProblemsEvaluate

2 2

3sin 2xe xdx

2xu e22 xdu e dx

cos 2dv xdx1 sin 22

v x

2 21 cos2 cos22

x xe x e xdx

21 cos 22

xe x

2 21 1sin 2 sin 2 22 2

x xe x x e dx

Practice ProblemsEvaluate

2 2

3sin 2xe xdx

2 21 cos2 sin 2 sin 2

2x xe x x e xdx

2 212 sin 2 cos 2 sin 22

x xe xdx e x x C

2 21sin 2 cos2 sin 24

x xe xdx e x x C Now, to apply the limits of integration…

Practice ProblemsEvaluate

2 2

3sin 2xe xdx

22

3

1 cos 2 sin 24

xe x x

4 61 1cos 4 sin 4 cos 6 sin 64 4e e

125.028 Verify numerically!!!