Integration by parts, just need a small hand

[SYoungblood]

Homework Statement

I'm going to cut from the initial part of the problem, which I am confident is good to go, and cut straight to the antiderivatives.

Homework Equations

All antiderivatives are to be integrated on the interval from 0 to π/18

(I₁) = -1/9 cos 9x - (I₂) (-2/27 * cos³(9x)) + (I₃) (-1/45 * cos⁵(9x)

The Attempt at a Solution

For I₁ -- (π)/18) -- -1/9 cos 9π/18 = -1/9 cos π/2 = 0
-1/9 cos 0 = -1/9
0 - (-1/9) = 1/9

ForI₂ -- ( π/18) -- -2/27 cos 9π/18 = -2/27 cos π/2 = 0
-2/27 cos 0 = -2/27
0 - (-2/27) = 2/27

For I₃ -- (π/18) -- -1/45 cos 9π/18 = -1/45 cos π/2 = 0
-1/45 cos 0 = -1/45
0- (-1/45) = 1/45

1/9 - 2/27 + 1/45 = 8/27, or 40/135 -- However, the answer in my text is 8/135.

Again, I am positive on my getting the antiderivatives right, I am sure the error is in the computing of the definite integrals. I thank you in advance for all help.

SY

 

[Ray Vickson]

Homework Statement

I'm going to cut from the initial part of the problem, which I am confident is good to go, and cut straight to the antiderivatives.

Homework Equations

All antiderivatives are to be integrated on the interval from 0 to π/18

(I₁) = -1/9 cos 9x - (I₂) (-2/27 * cos³(9x)) + (I₃) (-1/45 * cos⁵(9x)

The Attempt at a Solution

For I₁ -- (π)/18) -- -1/9 cos 9π/18 = -1/9 cos π/2 = 0
-1/9 cos 0 = -1/9
0 - (-1/9) = 1/9

ForI₂ -- ( π/18) -- -2/27 cos 9π/18 = -2/27 cos π/2 = 0
-2/27 cos 0 = -2/27
0 - (-2/27) = 2/27

For I₃ -- (π/18) -- -1/45 cos 9π/18 = -1/45 cos π/2 = 0
-1/45 cos 0 = -1/45
0- (-1/45) = 1/45

1/9 - 2/27 + 1/45 = 8/27, or 40/135 -- However, the answer in my text is 8/135.

Again, I am positive on my getting the antiderivatives right, I am sure the error is in the computing of the definite integrals. I thank you in advance for all help.

SY

Your individual terms are correct and your almost-answer of $1/9 - 2/27 + 1/45$ is correct, but your arithmetic is wrong after that.

 

[SYoungblood]

Your individual terms are correct and your almost-answer of $1/9 - 2/27 + 1/45$ is correct, but your arithmetic is wrong after that.

Got it, thank you very much for your help -- (15 - 10 + 3) / 135 = 8/ 135 -- just needed a push in the right direction.