Find an expression for y given that dy/dx = x²{x}

Correct Answer is : 2x727+c{2x^{7/2}}{7} + c72x27+c

Correct Answer is : 2x727+c{2x^{7/2}}{7} + c72x27+c

Test Index

WAEC Further Mathematics 2016 Paper

Question : 30

Total: 40

\frac{dy}{dx} = x^{2}\sqrt{x}

Solution:

Concept:

The given derivative is integrated using the power rule for functions of the form

x^n

Explanation:

Start with

\frac{dy}{dx} = x^{2}\sqrt{x}

Rewrite the square root as a fractional exponent:

x^{2}\sqrt{x} = x^{2} \cdot x^{\frac{1}{2}}

Add the exponents:

x^{2 + \frac{1}{2}} = x^{\frac{5}{2}}

Thus,

y = \int x^{\frac{5}{2}} \, dx

Apply

\int x^n \, dx = \frac{x^{n+1}}{n+1} + c

for

n = \frac{5}{2}

This gives

y = \frac{x^{\frac{5}{2}+1}}{\frac{5}{2}+1} + c

Since

\frac{5}{2}+1 = \frac{7}{2}

, we get

y = \frac{x^{\frac{7}{2}}}{\frac{7}{2}} + c

Dividing by

\frac{7}{2}

is the same as multiplying by

\frac{2}{7}

, so

y = \frac{2x^{\frac{7}{2}}}{7} + c

Answer:

y = \frac{2x^{\frac{7}{2}}}{7} + c

, which matches option C.

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