Solve the equation. (dy)/(dx)=(x^(2))/(9)+(7)/(9) Choose 1 answer: (A) y=(2x)/(9)+C (B) y=(x)/( 18)+C (C) y=(x^(3))/(3)+(7x)/(9)+C (D) y=(x^(3))/(27)+(7x)/(9)+C

Integrate with respect to x: Integrate both sides of the equation with respect to x. ∫(dy)/(dx) dx = ∫((x^(2))/(9) + (7)/(9)) dx y = ∫(x^(2)/9) dx + ∫(7/9) dxCalculate integral of x^2/9: Calculate the integral of x^2/9. ∫(x^(2)/9) dx = (1/9)∫x^2 dx = (1/9)(x^(3)/3) = (x^(3))/27Calculate integral of 7/9: Calculate the integral of 7/9. ∫(7/9) dx = (7/9)xCombine and add constant: Combine the results and add the constant of integration C. y = (x^(3))/27 + (7/9)x + C

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Calculus

Find derivatives of using multiple formulae

Full solution

Q. Solve the equation.

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\frac{d y}{d x}=\frac{x^{2}}{9}+\frac{7}{9}

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Choose

1

answer:

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(A)

y=\frac{2 x}{9}+C

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(B)

y=\frac{x}{18}+C

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(C)

y=\frac{x^{3}}{3}+\frac{7 x}{9}+C

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(D)

y=\frac{x^{3}}{27}+\frac{7 x}{9}+C

Integrate with respect to $x$ : Integrate both sides of the equation with respect to $x$ . $\int \frac{dy}{dx} dx = \int \left(\frac{x^{2}}{9} + \frac{7}{9}\right) dx$ $y = \int \left(\frac{x^{2}}{9}\right) dx + \int \left(\frac{7}{9}\right) dx$
Calculate integral of $x^2/9$ : Calculate the integral of $x^2/9$ . $\int(x^{2}/9) dx = (1/9)\int x^2 dx = (1/9)(x^{3}/3) = (x^{3})/27$
Calculate integral of $\frac{7}{9}$ : Calculate the integral of $\frac{7}{9}$ . $\int(\frac{7}{9}) \, dx = (\frac{7}{9})x$
Combine and add constant: Combine the results and add the constant of integration $C$ . $y = \frac{x^{3}}{27} + \frac{7}{9}x + C$