Two Questions on Integrals?

I've been trying to integrate these two functions forever, and my mind feels like it's been completely fried. Would anyone be able to explain the process behind the integrations? Thanks for any help...=)

1. Find F(x) if f(x) = y*sqrt(7+y)

2. Find F(x) if f(x) = t/(9+6t^2)

Again, thanks for your help.

1 Answer

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  • 1 decade ago
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    ∫ y √(7 + y) dy

    let u² = 7 + y ; 2u du = dy

    = 2 ∫ u² (u² - 7) du

    = 2/15 u³ (3u² - 35) + c

    = 2/15 (7 + y)^(3/2) (3y - 14) + c

    Answer (1): 2/15 (7 + y)^(3/2) (3y - 14) + c

    ∫ t / (9 + 6t²) dt

    let u = 9 + 6t² ; du = 12t dt

    = 1/12 ∫ du / u

    = 1/12 ln|u| + c

    = 1/12 ln(9 + 6t²) + c

    = 1/12 ln(3 + 2t²) + c + ln(3)

    = 1/12 ln(3 + 2t²) + c ......... that step is called "lumping constants"

    Answer (2): 1/12 ln(3 + 2t²) + c

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