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Math Help - Integration, Differential Equation and Determinant of a Matrix

  1. #1
    Junior Member
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    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
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  2. #2
    Super Member angel.white's Avatar
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    Quote Originally Posted by Confuzzled? View Post
    Hi there,

    Urgently need help with the following questions!
    1.Integral of cos^8xsin^3xdx
    \begin{gathered}<br />
   = \int {} cos^8 (x)sin^2 (x)sin(x)dx \hfill \\<br />
   = \int {} cos^8 (x)(1 - \cos ^2 (x))sin(x)dx \hfill \\ <br />
\end{gathered}<br />

    \begin{array}{|c|}<br />
\hline  SUBSTITUTION\\<br />
\hline  a = \cos (x) \\<br />
a~da =  - \sin (x)dx \\<br />
- a~da = \sin (x)dx \\\hline<br />
\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
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  3. #3
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    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 . . .

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