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8 (a) Given that $u = 2^x$, write down an expression for \( \frac{du}{dx} \) 8 (b) Find the exact value of \( \int_{2}^{3} \sqrt{3 + 2x} \, dx \) Fully justify your answer. - AQA - A-Level Maths: Pure - Question 8 - 2017 - Paper 1
Question 8
8 (a) Given that $u = 2^x$, write down an expression for \( \frac{du}{dx} \) 8 (b) Find the exact value of \( \int_{2}^{3} \sqrt{3 + 2x} \, dx \) Fully justify ... show full transcript
Worked Solution & Example Answer:8 (a) Given that $u = 2^x$, write down an expression for \( \frac{du}{dx} \) 8 (b) Find the exact value of \( \int_{2}^{3} \sqrt{3 + 2x} \, dx \) Fully justify your answer. - AQA - A-Level Maths: Pure - Question 8 - 2017 - Paper 1
Step 1
Step 2
Find the exact value of \( \int_{2}^{3} \sqrt{3 + 2x} \, dx \)
Answer
To solve the integral ( I = \int_{2}^{3} \sqrt{3 + 2x} , dx ), we start by simplifying the integrand. Let:
-
Substitution:
Let ( u = 3 + 2x ) Thus, ( \frac{du}{dx} = 2 ) or ( dx = \frac{du}{2} )
Changing the limits:
- When ( x = 2, \ u = 3 + 4 = 7 )
- When ( x = 3, \ u = 3 + 6 = 9 )
Now substituting:
- Integrate:
We have:
Evaluating at the limits: Since ( 9^{3/2} = 27 ) and ( 7^{3/2} = 7 \sqrt{7} $$:
Hence, the value of the integral is: