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Find $$\int (12x^6 - 3x^2 + 4x^1) \: dx$$ giving each term in its simplest form. - Edexcel - A-Level Maths Pure - Question 4 - 2011 - Paper 2

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Find--$$\int-(12x^6---3x^2-+-4x^1)-\:-dx$$--giving-each-term-in-its-simplest-form.-Edexcel-A-Level Maths Pure-Question 4-2011-Paper 2.png

Find $$\int (12x^6 - 3x^2 + 4x^1) \: dx$$ giving each term in its simplest form.

Worked Solution & Example Answer:Find $$\int (12x^6 - 3x^2 + 4x^1) \: dx$$ giving each term in its simplest form. - Edexcel - A-Level Maths Pure - Question 4 - 2011 - Paper 2

Step 1

Find $$\int (12x^6) \, dx$$

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Answer

The integral of 12x612x^6 is given by applying the power rule:

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

∫(12x6) dx=12⋅x6+16+1=2x7+C.\int (12x^6) \, dx = 12 \cdot \frac{x^{6+1}}{6+1} = 2x^7 + C.

Step 2

Find $$\int (-3x^2) \, dx$$

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Answer

Using the power rule again:

∫(−3x2) dx=−3⋅x2+12+1=−x3+C.\int (-3x^2) \, dx = -3 \cdot \frac{x^{2+1}}{2+1} = -x^3 + C.

Step 3

Find $$\int (4x^1) \, dx$$

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Answer

Again applying the power rule:

∫(4x1) dx=4⋅x1+11+1=2x2+C.\int (4x^1) \, dx = 4 \cdot \frac{x^{1+1}}{1+1} = 2x^2 + C.

Step 4

Combine all terms and add constant

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Answer

Now, combining all the results:

∫(12x6−3x2+4x1) dx=2x7−x3+2x2+C.\int (12x^6 - 3x^2 + 4x^1) \, dx = 2x^7 - x^3 + 2x^2 + C.

This gives each term in its simplest form.

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