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Find \( \int (2x^3 - 6x^{\frac{1}{2}}) \, dx, \, x \geq 0 - Scottish Highers Maths - Question 6 - 2023

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Question 6

Find-\(-\int-(2x^3---6x^{\frac{1}{2}})-\,-dx,-\,-x-\geq-0-Scottish Highers Maths-Question 6-2023.png

Find \( \int (2x^3 - 6x^{\frac{1}{2}}) \, dx, \, x \geq 0. \)

Worked Solution & Example Answer:Find \( \int (2x^3 - 6x^{\frac{1}{2}}) \, dx, \, x \geq 0 - Scottish Highers Maths - Question 6 - 2023

Step 1

Express second term in integrable form

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Answer

Rewrite the integrand: ( 2x^3 - 6x^{\frac{1}{2}} ).

Step 2

Integrate one term

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Answer

Integrate the first term: ( \int 2x^3 , dx = \frac{2}{4} x^4 = \frac{1}{2} x^4 + C. )

Step 3

Integrate other term

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Answer

Integrate the second term: ( \int -6x^{\frac{1}{2}} , dx = -6 \cdot \frac{2}{3} x^{\frac{3}{2}} = -4x^{\frac{3}{2}}. )

Step 4

Complete integration

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Answer

Combine the results: ( \int (2x^3 - 6x^{\frac{1}{2}}) , dx = \frac{1}{2} x^4 - 4x^{\frac{3}{2}} + C. )

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