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  1. Top | #25

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    \[Evaluate:\,\,\int {\sqrt {\frac{{1 + \sin 8x}}{{1 - \sin 8x}}} } \,\,dx\]


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  3. Top | #26

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  5. Top | #27

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    [URL="http://alnasiry.net/forums"][IMG]http://alnasiry.net/forums/uploaded/2_iraqiflag.gif[/IMG][/URL]


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  7. Top | #28

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  9. Top | #29

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    \[\int\limits_0^{\pi /2} {\frac{2}{{{{\cos }^2}\theta + 4{{\sin }^2}\theta }}\,\,d\theta } \,\,\]

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  13. Top | #31

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    \[Evaluate:\int\limits_0^\infty {{x^2}{e^{ - x}}} \,dx\]

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  15. Top | #32

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  17. Top | #33

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    \[\begin{array}{l}
    \left( 1 \right)\,\,\,\,\int_0^a {f\left( x \right)} \,dx = \,\int_0^a {f\left( {a - x} \right)} \,dx\\
    \left( 2 \right)\,\,\,\,\int_a^b {f\left( x \right)} \,dx = \,\int_a^b {f\left( {a + b - x} \right)} \,dx\\
    \left( 3 \right)\,\,\,\,\int_{ - a}^a {f\left( {{x^2}} \right)} \,dx = \,2\int_0^a {f\left( {{x^2}} \right)} \,dx\\
    \left( 4 \right)\,\,\,\,\int_0^{\pi /2} {{{\sin }^2}x} \,dx = \,\,\int_0^{\pi /2} {{{\cos }^2}x} \,dx
    \end{array}\]












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  19. Top | #34

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    ****


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  23. Top | #36

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    \[\begin{array}{l}
    \left( 2 \right)\,\,\,\int\limits_a^b {f\left( {a + b - x} \right)} \,\,dx = \int\limits_a^b {f\left( x \right)} \,\,dx\\
    The\,\,\Pr ove\,:\\
    Putting\,\,a + b - x = y\,\,\,then\,\,\, - dx = dy\,\,\,,\,\\
    \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,wh en\,\,x = a \Rightarrow y = b\,\,and\,\,\,when\,\,x = b \Rightarrow y = a\\
    \,\,\,\int\limits_a^b {f\left( {a + b - x} \right)} \,\,dx = \,\int\limits_b^a {f\left( y \right)} \,\left( {\, - dy} \right) = \underbrace {\int\limits_a^b {f\left( y \right)} \,\,dy = \int\limits_a^b {f\left( x \right)} \,\,dx}_{Changing\,y\,to\,x}
    \end{array}\]


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