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Find \( \int (12x^3 - 8x^2 + 3) \, dx \), giving each term in its simplest form. - Edexcel - A-Level Maths Pure - Question 4 - 2009 - Paper 1

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Find-\(-\int-(12x^3---8x^2-+-3)-\,-dx-\),-giving-each-term-in-its-simplest-form.-Edexcel-A-Level Maths Pure-Question 4-2009-Paper 1.png

Find \( \int (12x^3 - 8x^2 + 3) \, dx \), giving each term in its simplest form.

Worked Solution & Example Answer:Find \( \int (12x^3 - 8x^2 + 3) \, dx \), giving each term in its simplest form. - Edexcel - A-Level Maths Pure - Question 4 - 2009 - Paper 1

Step 1

Evaluate \( \int (12x^3) \, dx \)

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Answer

To find the integral of ( 12x^3 ), we apply the power rule of integration:

∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C

Thus,

∫12x3 dx=12⋅x44=3x4+C.\int 12x^3 \, dx = 12 \cdot \frac{x^{4}}{4} = 3x^{4} + C.

Step 2

Evaluate \( \int (-8x^2) \, dx \)

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Answer

Next, we integrate ( -8x^2 ) using the same power rule:

∫−8x2 dx=−8⋅x33=−83x3+C.\int -8x^2 \, dx = -8 \cdot \frac{x^{3}}{3} = -\frac{8}{3}x^{3} + C.

Step 3

Evaluate \( \int (3) \, dx \)

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Answer

Finally, for ( 3 ), the integral is simply:

∫3 dx=3x+C.\int 3 \, dx = 3x + C.

Step 4

Combine the results

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Answer

The final expression for the integral is the sum of all parts:

∫(12x3−8x2+3) dx=3x4−83x3+3x+C,\int (12x^3 - 8x^2 + 3) \, dx = 3x^{4} - \frac{8}{3}x^{3} + 3x + C, where ( C ) is the constant of integration.

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