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TomLikesPhysics

  • 2 years ago

I need help with an Integral (Substitution).

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  1. TomLikesPhysics
    • 2 years ago
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    \[\int\limits_{0}^{100}e^\sqrt{x}dx=?\] I am supposed to solve this using the method of substitution. I have no clue how the hell this would help in that weird case.

  2. hartnn
    • 2 years ago
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    u know integration by parts ?

  3. across
    • 2 years ago
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    Let\[ u=\sqrt x.\tag{1}\]Then,\[ du=\frac1{2\sqrt x}dx\Longrightarrow dx=2\sqrt xdu=2udu.\tag{2} \]It follows from \((1)\) that\[ x=0\Longrightarrow u=0\text{, and}\\ x=100\Longrightarrow u=10. \]So, from \((1)\), \((2)\) and the new limits, we have that\[ 2\int_0^{10}ue^u\,du. \]

  4. TomLikesPhysics
    • 2 years ago
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    @hartnn Yes, that would be my first impulse but I should use the substitution method.

  5. hartnn
    • 2 years ago
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    as shown by @across

  6. TomLikesPhysics
    • 2 years ago
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    Ok, and after that I have to integrate by parts do integrate the \[ue^udu\] part, right?

  7. hartnn
    • 2 years ago
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    yes.

  8. across
    • 2 years ago
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    Yes, or you could use the (not so) common knowledge that\[ \int xe^x\,dx=e^x(x-1). \]

  9. TomLikesPhysics
    • 2 years ago
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    ^^ I thought that I saw that Integral before, but not today it seems. Thx for your help.

  10. hartnn
    • 2 years ago
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    more general formula : \(\large \int e^x(f(x)+f'(x))dx=e^xf(x)+c\)

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