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Math Help - convergent integral

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    convergent integral



    so im pretty sure its convergent but the answer i get is arctan(6) anyone know what im doing wrong.
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    Quote Originally Posted by dat1611 View Post


    so im pretty sure its convergent but the answer i get is arctan(6) anyone know what im doing wrong.
    \int\limits_{ - \infty }^6 {\frac{{dx}}{{{x^2} + 1}}}  = \left. {\arctan x} \right|_{ - \infty }^6 = \arctan 6 - \underbrace {\mathop {\lim }\limits_{x \to  - \infty } \arctan x}_{ - \pi /2} = \arctan 6 + \frac{\pi }{2}.
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    Quote Originally Posted by dat1611 View Post


    so im pretty sure its convergent but the answer i get is arctan(6) anyone know what im doing wrong.
    the integral is convergent ...

    \lim_{a \to -\infty} \int_a^6 \frac{1}{x^2+1} \, dx

    \lim_{a \to -\infty} \left[\arctan{x}\right]_a^6

    \lim_{a \to -\infty} \left[\arctan(6) - \arctan{a}\right] = \arctan(6) - \left(-\frac{\pi}{2}\right)  = \arctan(6) + \frac{\pi}{2}
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