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Find \[ \int (8x^3 + 6x^2 - 5) \, dx \] giving each term in its simplest form. - Edexcel - A-Level Maths Pure - Question 4 - 2010 - Paper 1

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Find-\[-\int-(8x^3-+-6x^2---5)-\,-dx-\]-giving-each-term-in-its-simplest-form.-Edexcel-A-Level Maths Pure-Question 4-2010-Paper 1.png

Find \[ \int (8x^3 + 6x^2 - 5) \, dx \] giving each term in its simplest form.

Worked Solution & Example Answer:Find \[ \int (8x^3 + 6x^2 - 5) \, dx \] giving each term in its simplest form. - Edexcel - A-Level Maths Pure - Question 4 - 2010 - Paper 1

Step 1

Find \( \int 8x^3 \, dx \)

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Answer

To integrate ( 8x^3 ), we use the power rule of integration, which states that ( \int x^n , dx = \frac{x^{n+1}}{n+1} + C ). Therefore,\n[ \int 8x^3 , dx = 8 \cdot \frac{x^{4}}{4} = 2x^{4} ]

Step 2

Find \( \int 6x^2 \, dx \)

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Answer

Applying the power rule again,\n[ \int 6x^2 , dx = 6 \cdot \frac{x^{3}}{3} = 2x^{3} ]

Step 3

Find \( \int -5 \, dx \)

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Answer

The integral of a constant is just the constant multiplied by ( x ):\n[ \int -5 , dx = -5x ]

Step 4

Combine all parts

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Answer

Now, we combine all the integrals: \n[ \int (8x^3 + 6x^2 - 5) , dx = 2x^4 + 2x^3 - 5x + C ] where ( C ) is the constant of integration.

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