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Math Help - Find the integral

  1. #1
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    Find the integral

    Hello, i need help with this problem, here's what i'm doing.
    (6x-7)^1/2 dx

    would it not be 3/2(6x-7)^3/2
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  2. #2
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    No.

    To make it more clear for you, sub u = 6x-7, du = 6 dx. Therefore:

    \int (6x-7)^{\frac{1}{2}}~dx = \frac{1}{6} \int (u)^{\frac{1}{2}}~du = \frac{u^{\frac{3}{2}}}{9} + C
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  3. #3
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    Hello, plstevens!

    Hello, i need help with this problem.

    Here's what i'm doing: . \int (6x-7)^{\frac{1}{2}}\,dx

    would it not be: . \frac{3}{2}(6x-7)^{\frac{3}{2}} . . . . no
    At least three errors . . .

    Let u \:= \:6x-7 \quad\Rightarrow\quad du \:=\:6\,dx \quad\Rightarrow\quad dx \:=\:\tfrac{1}{6}\,du

    Substitute: . \int u^{\frac{1}{2}}\left(\tfrac{1}{6}\,du\right) \;=\;\tfrac{1}{6}\int u^{\frac{1}{2}}\,du \;=\; \frac{\frac{1}{6}}{\frac{3}{2}}\,u^{\frac{3}{2}} + C \;=\;\tfrac{1}{9}u^{\frac{3}{2}} + C

    Back-substitute: . \tfrac{1}{9}(6x-7)^{\frac{3}{2}} + C

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  4. #4
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    Solution

    Well, you would start by letting u = 6x - 7
    Take the derivative of both sides to get:
    du = 6dx; Solve for dx to get: dx = du/6. Now substitute this into the original integral to get:

    ∫[(u)^(1/2)]du/6

    Note, u^1/2 is the same as sqrt(u)

    Now, because 1/6 is a constant you can put it in front of the integral.

    (1/6)∫[u^(1/2)]du

    By using the reverse power rule, the integration would result in:
    (1/6)*(2/3)*(u^3/2) + C, for some constant C.

    Reduce the fractions and substitute the u value back in to yield:

    (1/9)*[(6x-7)^3/2] + C.

    Note: Since the integral is indefinite, you must include the constant C. Teachers will always take off points on tests if you forget this. I hope this helps. Also, you could take the derivative of this to check that you get (6x-7)^(1/2)
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