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8 (a) Given that $u = 2^x$, write down an expression for \( \frac{du}{dx} \) - 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 \( \int_{0}^{2} \sqrt{3 + 2x} \, dx \) Fully justify yo... 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 Pure - Question 8 - 2017 - Paper 1
Step 1
Step 2
Find the exact value of \( \int_{0}^{2} \sqrt{3 + 2x} \, dx \)
Answer
First, we make a substitution to simplify the integral. Let:
Then:
ightarrow dx = \frac{dt}{2}$$ Next, we will change the limits of integration. When \( x = 0 \): $$t = 3 + 2(0) = 3$$ When \( x = 2 \): $$t = 3 + 2(2) = 7$$ Now we can rewrite the integral: $$I = \int_{3}^{7} \sqrt{t} \cdot \frac{1}{2} \, dt$$ This simplifies to: $$I = \frac{1}{2} \int_{3}^{7} t^{\frac{1}{2}} \, dt$$ We can now integrate \( t^{\frac{1}{2}} \): $$= \frac{1}{2} \left[ \frac{t^{\frac{3}{2}}}{\frac{3}{2}} \right]_3^7$$ This gives us: $$= \frac{1}{3} \left[ t^{\frac{3}{2}} \right]_3^7 = \frac{1}{3} \left( 7^{\frac{3}{2}} - 3^{\frac{3}{2}} \right)$$ Finally, calculating the values: \( 7^{\frac{3}{2}} = 7 \sqrt{7} \) and \( 3^{\frac{3}{2}} = 3 \sqrt{3} \), so: $$= \frac{1}{3} (7 \sqrt{7} - 3 \sqrt{3})$$