# Thread: Integration, Differential Equation and Determinant of a Matrix

1. ## 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

2. Originally Posted by Confuzzled?
Hi there,

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

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

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

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

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

Unsubstitute
$= \frac {cos^9(x)}{9} - \frac{cos^{11}(x)}{11}+C$

3. Hello, Confuzzled!

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

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

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

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

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

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

I'll let you simplify it . . .