# Integration by Parts

• May 30th 2007, 07:16 PM
asnxbbyx113
Integration by Parts
I need help with 4 of these. I tried them, but I keep getting the wrong answer.

1. Evalute the integral of (sin(8x))^3*(cos(8x))^2*dx. Use C to denote an arbitrary constant.

2. Evalute the integral of (cos(5x))^3*dx. Use C to denote an arbitrary constant.

3. Evalute the integral of csc(3x)dx. Use C to denote an arbitrary constant.

4. Evalute the integral of (2-2sinx)/cosx dx. Use C to denote an arbitrary constant.
• May 30th 2007, 07:18 PM
ThePerfectHacker
Quote:

Originally Posted by asnxbbyx113
4. Evalute the integral of (2-2sinx)/cosx dx. Use C to denote an arbitrary constant.

Hint:

$\displaystyle \int \frac{2-2\sin x}{\cos x} dx = \int \frac{2}{\cos x} - \frac{2\sin x}{\cos x} dx$

$\displaystyle = \int 2\sec x - 2\tan x dx$
• May 30th 2007, 07:20 PM
ThePerfectHacker
Quote:

Originally Posted by asnxbbyx113

3. Evalute the integral of csc(3x)dx. Use C to denote an arbitrary constant.
.

$\displaystyle \int \csc 3x dx$

Let $\displaystyle t=3x$:

$\displaystyle \frac{1}{3} \int \csc t dt$

$\displaystyle =\frac{1}{3} \left( \ln|\csc x - \cot x| + C\right)$
• May 30th 2007, 07:23 PM
ThePerfectHacker
Quote:

Originally Posted by asnxbbyx113
I need help with 4 of these. I tried them, but I keep getting the wrong answer.

1. Evalute the integral of (sin(8x))^3*(cos(8x))^2*dx. Use C to denote an arbitrary constant.

2. Evalute the integral of (cos(5x))^3*dx. Use C to denote an arbitrary constant.
.

$\displaystyle \int \cos^3 x dx = \int \cos x \cos^2 x dx = \int \cos x (1-\sin^2 x) dx$

Let $\displaystyle t=\sin x$ then $\displaystyle t'=\cos x$:

$\displaystyle \int (1-t^2)dt = t - \frac{1}{3}t^3+C$

$\displaystyle \sin x - \frac{1}{3}\sin^3 x +C$

Now to find,

$\displaystyle \int \cos^3 5x dx$

Just use the substitution $\displaystyle u=5x$ und do what I do above.
• May 30th 2007, 07:25 PM
ThePerfectHacker
Quote:

Originally Posted by asnxbbyx113
I need help with 4 of these. I tried them, but I keep getting the wrong answer.

1. Evalute the integral of (sin(8x))^3*(cos(8x))^2*dx. Use C to denote an arbitrary constant.

$\displaystyle \int \sin^3 8x \cos^2 8x dx$

$\displaystyle =\frac{1}{8} \int \sin^3 u \cos^2 u du$

$\displaystyle =\frac{1}{8} \int \sin^2 u \cos^2 u \sin u du$

$\displaystyle =\frac{1}{8} \int (1-\cos^2 u)\cos^2 u \sin u du$

Let $\displaystyle t=\cos u$ then,

$\displaystyle -\frac{1}{8} \int (1-t^2)t^2 dt$

You finish!