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

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(02-01-2019)

4. :

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

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6. The Following User Says Thank You to For This Useful Post:

(02-03-2019)

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(02-03-2019)

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(02-03-2019)

10. :

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

11. :

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

Q1 kamil.jpg

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(02-20-2019)

13. :

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

14. :

(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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(02-26-2019)

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