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Math Help - Improper Integrals

  1. #1
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    Improper Integrals

    I can't figure out this one...

    Let f(x) =  \frac{x^2 + 2}{x^2 + 1}

    a) Show that f(x) > 1 for all x

    b) Use the inequality in part (a) to show that  \int_{-\infty}^{\infty}f(x)dx diverges.

    Do I start with L' Hopitals?
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  2. #2
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    Quote Originally Posted by larson View Post
    I can't figure out this one...

    Let f(x) =  \frac{x^2 + 2}{x^2 + 1}

    a) Show that f(x) > 1 for all x

    b) Use the inequality in part (a) to show that  \int_{-\infty}^{\infty}f(x)dx diverges.

    Do I start with L' Hopitals?
    (a)  \frac{x^2 + 2}{x^2 + 1} = \frac{(x^2 + 1) + 1}{x^2 + 1} = 1 + \frac{1}{x^2 + 1} .....


    (b)  \int_{-\infty}^{\infty}f(x) \, dx >  \int_{-\infty}^{\infty}1 \, dx .......
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by mr fantastic View Post
    (a)  \frac{x^2 + 2}{x^2 + 1} = \frac{(x^2 + 1) + 1}{x^2 + 1} = 1 + \frac{1}{x^2 + 1} .....


    (b)  \int_{-\infty}^{\infty}f(x) \, dx > \int_{-\infty}^{\infty}1 \, dx .......
    as Mr. Fantastic showed you you can rewrite his as \int_{-\infty}^{\infty}1+\frac{1}{x^2+1}dx...which is \bigg[x+arctan(x)\bigg]\bigg|_{-\infty}^{\infty}=\bigg[{\infty}+\frac{\pi}{2}\bigg]-\bigg[-{\infty}-\frac{\pi}{2}\bigg]\Rightarrow{divergent}
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