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  1. Top | #13

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  2. Top | #14

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    6) 1/2e^4
    8) o
    10) 0

  3. Top | #15

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  5. Top | #16

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    \[\begin{array}{l}
    Q8\\
    \int\limits_{ - \infty }^\infty {\frac{x}{{\sqrt {{x^2} + 2} }}} \,\,dx\\
    Solution:\int\limits_{ - \infty }^\infty {\frac{x}{{\sqrt {{x^2} + 2} }}} \,\,dx = \int\limits_{ - \infty }^0 {\frac{x}{{\sqrt {{x^2} + 2} }}} \,dx + \int\limits_0^\infty {\frac{x}{{\sqrt {{x^2} + 2} }}} \,\,dx\\
    1)\,\,\int\limits_{ - \infty }^0 {\frac{x}{{\sqrt {{x^2} + 2} }}} \,\,dx = \mathop {\lim }\limits_{b \to - \infty } \int\limits_b^0 {\frac{x}{{\sqrt {{x^2} + 2} }}} \,\,dx = \mathop {\lim }\limits_{b \to - \infty } \left. {\sqrt {{x^2} + 2} } \right]\begin{array}{*{20}{c}}
    0\\
    b
    \end{array}\\
    \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{b \to - \infty } \left. {\left( {\sqrt 2 - \sqrt {{b^2} + 2} } \right)} \right] = \sqrt 2 - \infty = - \infty \,\,\\
    2)In\,same\,way\,\,\,\,\int\limits_0^\infty {\frac{x}{{\sqrt {{x^2} + 2} }}} \,\,dx = \mathop {\lim }\limits_{a \to \infty } \int\limits_0^a {\frac{x}{{\sqrt {{x^2} + 2} }}} \,\,dx = \mathop {\lim }\limits_{a \to \infty } \left. {\sqrt {{x^2} + 2} } \right]\begin{array}{*{20}{c}}
    a\\
    0
    \end{array}\\
    \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{a \to \infty } \left. {\left( {\sqrt {{a^2} + 2} - \sqrt 2 } \right)} \right] = \infty - \sqrt 2 = \infty \,\,\\
    Hence\,\,\int\limits_{ - \infty }^\infty {\frac{x}{{\sqrt {{x^2} + 2} }}} \,\,dx\,\,\,is\,diverges
    \end{array}\]










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  7. Top | #17

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  10. Top | #19

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  12. Top | #20

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  14. Top | #21

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    \[\begin{array}{l}
    Evaluate:\\
    \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 1 \right)\,\,\int {{{\tan }^{ - 1}}\left( {\frac{{\sin x}}{{1 + \cos x}}} \right)} \,dx\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,K1
    \end{array}\]



    ; 02-20-2019 01:13 AM

  15. Top | #22

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    Tan (x/2)=sin x/(1+cos x) implies the integral=integral of x/2=1/4x^2 +c




    Q1 kamil.jpg
    ; 02-24-2019 12:52 AM

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  17. Top | #23

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    \[\left( 2 \right)\,\,\int {{{\tan }^{ - 1}}\left( {\frac{{\cos x}}{{1 + \sin x}}} \right)} \,dx\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,K1\]

    ; 02-24-2019 12:47 AM

  18. Top | #24

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    (Tan (pi/4+x/2)=cos x/(1+sin x)
    =then the integral= integral of pi/4+x/2
    Pi/4 x+x^2/4+c

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