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{:[G(x)=(x+3)^(2)],[g(x)=G^(')(x)],[int_(-1)^(7)g(x)dx=]:}

G(x)=(x+3)2g(x)=G′(x)∫−17g(x)dx= \begin{array}{l}G(x)=(x+3)^{2} \\ g(x)=G^{\prime}(x) \\ \int_{-1}^{7} g(x) d x=\end{array}

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Find higher derivatives of rational and radical functionsright chevron icon

Full solution

Q. G(x)=(x+3)2g(x)=G′(x)∫−17g(x)dx= \begin{array}{l}G(x)=(x+3)^{2} \\ g(x)=G^{\prime}(x) \\ \int_{-1}^{7} g(x) d x=\end{array}
  1. Find Derivative of G(x): First, we need to find the derivative of G(x), which is g(x)g(x).g(x)=G′(x)=ddx[(x+3)2]g(x) = G'(x) = \frac{d}{dx}[(x+3)^2]Using the power rule, the derivative of (x+3)2(x+3)^2 is 2∗(x+3)2*(x+3).So, g(x)=2∗(x+3)g(x) = 2*(x+3)
  2. Calculate Integral of g(x)g(x): Now, we need to calculate the integral of g(x)g(x) from −1-1 to 77.
    ∫−172⋅(x+3) dx\int_{-1}^{7} 2\cdot(x+3) \, dx
    This is a simple polynomial integration problem.

    Integrate 2⋅(x+3)2\cdot(x+3) with respect to xx.
    The antiderivative of 2⋅(x+3)2\cdot(x+3) is 2⋅(12)⋅x2+2⋅3⋅x2\cdot(\frac{1}{2})\cdot x^2 + 2\cdot3\cdot x, which simplifies to x2+6xx^2 + 6x.
    So, g(x)g(x)00
  3. Evaluate Antiderivative: Evaluate the antiderivative from −1-1 to 77.\newlinePlug in the upper limit: (7)2+6∗(7)=49+42=91(7)^2 + 6*(7) = 49 + 42 = 91\newlinePlug in the lower limit: (−1)2+6∗(−1)=1−6=−5(-1)^2 + 6*(-1) = 1 - 6 = -5\newlineNow subtract the lower limit result from the upper limit result.\newline91−(−5)=91+5=9691 - (-5) = 91 + 5 = 96

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