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8 (a) Given that $u = 2^x$, write down an expression for \( \frac{du}{dx} \) - AQA - A-Level Maths Mechanics - 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 \( \int_0^{2} \sqrt{3 + 2x} \, dx \) Fully justify your... show full transcript
Worked Solution & Example Answer:8 (a) Given that $u = 2^x$, write down an expression for \( \frac{du}{dx} \) - AQA - A-Level Maths Mechanics - Question 8 - 2017 - Paper 1
Step 1
Step 2
Find the exact value of \( \int_0^{2} \sqrt{3 + 2x} \, dx \)
Answer
To solve the integral, we can perform a substitution. Let:
[ u = 3 + 2x ] This implies [ du = 2 , dx \quad \Rightarrow \quad dx = \frac{du}{2} ]
Next, we change the limits of integration:
- When ( x = 0, \ u = 3 + 2(0) = 3 )
- When ( x = 2, \ u = 3 + 2(2) = 7 )
Thus, the integral becomes:
[ I = \int_{3}^{7} \sqrt{u} \cdot \frac{du}{2} = \frac{1}{2} \int_{3}^{7} u^{1/2} , du ]
Now we integrate:
[ I = \frac{1}{2} \cdot \left[ \frac{u^{3/2}}{3/2} \right]{3}^{7} = \frac{1}{3} \left[ u^{3/2} \right]{3}^{7} ]
Calculating the bounds gives:
[ \frac{1}{3} \left[ 7^{3/2} - 3^{3/2} \right] = \frac{1}{3} \left[ 7\sqrt{7} - 3\sqrt{3} \right] ]
Thus, the exact value is:
[ I = \frac{7\sqrt{7} - 3\sqrt{3}}{3} ]