Find
\[ \int \left( 2x^4 - \frac{4}{\sqrt{x}} + 3 \right) dx \]
giving each term in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2016 - Paper 1
Question 3
Find
\[ \int \left( 2x^4 - \frac{4}{\sqrt{x}} + 3 \right) dx \]
giving each term in its simplest form.
Worked Solution & Example Answer:Find
\[ \int \left( 2x^4 - \frac{4}{\sqrt{x}} + 3 \right) dx \]
giving each term in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2016 - Paper 1
Step 1
Find \( \int (2x^4) dx \)
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Answer
To integrate ( 2x^4 ), use the power rule: [ \int x^n dx = \frac{x^{n+1}}{n+1} + C ]. Thus,[ \int 2x^4 dx = 2 \cdot \frac{x^{5}}{5} = \frac{2}{5} x^5 + C_1. ]
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Answer
Rewriting ( -\frac{4}{\sqrt{x}} ) as ( -4x^{-\frac{1}{2}} ), we apply the power rule again: [ \int -4x^{-\frac{1}{2}} dx = -4 \cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = -8x^{\frac{1}{2}} + C_2. ]
Step 3
Find \( \int 3 dx \)
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Answer
The integral of a constant is calculated as follows: [ \int 3 dx = 3x + C_3. ]
Step 4
Combine all terms
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Answer
Combining all the integrated results, we have: [ \int \left( 2x^4 - \frac{4}{\sqrt{x}} + 3 \right) dx = \frac{2}{5} x^5 - 8\sqrt{x} + 3x + C. ] In its simplest form, this is the final answer.
![Find-\[-\int-\left(-2x^4---\frac{4}{\sqrt{x}}-+-3-\right)-dx-\]-giving-each-term-in-its-simplest-form.-Edexcel-A-Level Maths Pure-Question 3-2016-Paper 1.png](https://cdn.simplestudy.io/assets/backend/uploads/question/MathsPure_2016_1_1_QP_1_2016 .jpg)
