Evaluate \( \int_1^8 \frac{1}{\sqrt{x}} \, dx \), giving your answer in the form \( a + b\sqrt{2} \), where \( a \) and \( b \) are integers. - Edexcel - A-Level Maths Pure - Question 3 - 2007 - Paper 2
Question 3
Evaluate \( \int_1^8 \frac{1}{\sqrt{x}} \, dx \), giving your answer in the form \( a + b\sqrt{2} \), where \( a \) and \( b \) are integers.
Worked Solution & Example Answer:Evaluate \( \int_1^8 \frac{1}{\sqrt{x}} \, dx \), giving your answer in the form \( a + b\sqrt{2} \), where \( a \) and \( b \) are integers. - Edexcel - A-Level Maths Pure - Question 3 - 2007 - Paper 2
Step 1
Evaluate the integral \( \int \frac{1}{\sqrt{x}} \, dx \)
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Answer
To solve ( \int \frac{1}{\sqrt{x}} , dx ), we can use the power rule of integration. The integral can be rewritten as:\n[ \int x^{-1/2} , dx = 2x^{1/2} + C ]\nThus, the indefinite integral is ( 2\sqrt{x} + C )
Step 2
Substituting the limits 1 and 8
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Answer
Now we will evaluate the definite integral from 1 to 8:
[ \int_1^8 \frac{1}{\sqrt{x}} , dx = \left[ 2\sqrt{x} \right]_1^8 = 2\sqrt{8} - 2\sqrt{1} ]\nEvaluating this gives:
[ 2\sqrt{8} - 2 = 2(2\sqrt{2}) - 2 = 4\sqrt{2} - 2 ]
Step 3
Final Answer
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Answer
The answer can be expressed in the required form ( a + b\sqrt{2} ):
[ -2 + 4\sqrt{2} ]\nThus, here ( a = -2 ) and ( b = 4 ).
