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Find \[ \int (10x^4 - 4x - \frac{3}{\sqrt{x}}) dx \] giving each term in its simplest form. - Edexcel - A-Level Maths Pure - Question 4 - 2013 - Paper 1

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Find-\[-\int-(10x^4---4x---\frac{3}{\sqrt{x}})-dx-\]-giving-each-term-in-its-simplest-form.-Edexcel-A-Level Maths Pure-Question 4-2013-Paper 1.png

Find \[ \int (10x^4 - 4x - \frac{3}{\sqrt{x}}) dx \] giving each term in its simplest form.

Worked Solution & Example Answer:Find \[ \int (10x^4 - 4x - \frac{3}{\sqrt{x}}) dx \] giving each term in its simplest form. - Edexcel - A-Level Maths Pure - Question 4 - 2013 - Paper 1

Step 1

Evaluate \( \int 10x^4 \, dx \)

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Answer

To integrate ( 10x^4 ), we apply the power rule for integration: [ \int 10x^4 , dx = 10 \cdot \frac{x^{5}}{5} = 2x^{5} ]

Step 2

Evaluate \( \int -4x \, dx \)

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Answer

Using the power rule again: [ \int -4x , dx = -4 \cdot \frac{x^{2}}{2} = -2x^{2} ]

Step 3

Evaluate \( \int -\frac{3}{\sqrt{x}} \, dx \)

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Answer

We rewrite ( \frac{1}{\sqrt{x}} ) as ( x^{-\frac{1}{2}} ) and integrate: [ \int -3x^{-\frac{1}{2}} , dx = -3 \cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = -6x^{\frac{1}{2}} ]

Step 4

Combine the results and simplify

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Answer

Putting it all together, we have: [ \int (10x^4 - 4x - \frac{3}{\sqrt{x}}) , dx = 2x^{5} - 2x^{2} - 6x^{\frac{1}{2}} + C ] where ( C ) is the constant of integration.

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