# Integration, Differential Equation and Determinant of a Matrix

• Feb 17th 2008, 08:58 AM
Confuzzled?
Integration, Differential Equation and Determinant of a Matrix
Hi there,

Urgently need help with the following questions!
1.Integral of cos^8xsin^3xdx
2.Integral of (x^2 +6x)dy=y^2dx-12dy
3.Determinant of:
3-λ -1 2
5 -3-λ 5
1 -1 2-λ

Thanks so much if you can help! xxx
• Feb 17th 2008, 09:30 AM
angel.white
Quote:

Originally Posted by Confuzzled?
Hi there,

Urgently need help with the following questions!
1.Integral of cos^8xsin^3xdx

$\displaystyle \begin{gathered} = \int {} cos^8 (x)sin^2 (x)sin(x)dx \hfill \\ = \int {} cos^8 (x)(1 - \cos ^2 (x))sin(x)dx \hfill \\ \end{gathered}$

$\displaystyle \begin{array}{|c|} \hline SUBSTITUTION\\ \hline a = \cos (x) \\ a~da = - \sin (x)dx \\ - a~da = \sin (x)dx \\\hline \end{array}$

$\displaystyle = \int a^8(1 - a^2)da$

$\displaystyle = \int (a^8 - a^{10})da$

$\displaystyle = \frac {a^9}{9} - \frac{a^{11}}{11}+C$

Unsubstitute
$\displaystyle = \frac {cos^9(x)}{9} - \frac{cos^{11}(x)}{11}+C$
• Feb 17th 2008, 10:07 AM
Soroban
Hello, Confuzzled!

Quote:

2. Integral of $\displaystyle (x^2 +6x)dy \:=\:y^2dx-12dy$

We have: .$\displaystyle (x^2 + 6x)dy + 12dy \:=\:y^2dx$

. . . . . . . . . $\displaystyle (x^2 + 6x + 12)dy \;=\;y^2dx$

Separate variables: .$\displaystyle \frac{dy}{y^2} \;=\;\frac{dx}{x^2+6x+12}$

Integrate: .$\displaystyle \int y^{-2}dy\;=\;\int\frac{dx}{(x+3)^2 + (\sqrt{3})^2}$

. . and we have: .$\displaystyle -y^{-1} \;=\;\frac{1}{\sqrt{3}}\arctan\left(\frac{x+3}{\sq rt{3}}\right) + C$

I'll let you simplify it . . .