A-Level AQA Maths Pure Further Integration: Use integration by substitution

Use integration by substitution to show that $$\int_{-4}^{6} \sqrt{4x+1} \: dx = \frac{875}{12}$$ Fully justify your answer. - AQA - A-Level Maths Pure - Question 5 - 2020 - Paper 2

Question 5

$Use-integration-by-substitution-to-show-that--$$\int_{-4}^{6}-\sqrt{4x+1}-\:-dx-=-\frac{875}{12}$$--Fully-justify-your-answer.-AQA-A-Level Maths Pure-Question 5-2020-Paper 2.png$

Use integration by substitution to show that $$\int_{-4}^{6} \sqrt{4x+1} \: dx = \frac{875}{12}$$ Fully justify your answer.

Worked Solution & Example Answer:Use integration by substitution to show that $$\int_{-4}^{6} \sqrt{4x+1} \: dx = \frac{875}{12}$$ Fully justify your answer. - AQA - A-Level Maths Pure - Question 5 - 2020 - Paper 2

Step 1

Use a suitable substitution

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Answer

Let us choose the substitution:

$u = 4x + 1.$
Then, the differential is:

$du = 4 \: dx, \quad \text{or} \quad dx = \frac{du}{4}.$
We also need to change the limits of integration:

When $x = -4$ , $u = 4(-4) + 1 = -15$ ;
When $x = 6$ , $u = 4(6) + 1 = 25.$

Step 2

Completes substitution to obtain correct integrand

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Answer

Substituting into the integral, we get:

$$\int_{-15}^{25} \sqrt{u} : \frac{du}{4} = \frac{1}{4} \int_{-15}^{25} u^{1/2} : du.$

Step 3

Correctly integrates their simplified integrand

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Answer

Now, we can integrate:

$\frac{1}{4} \int u^{1/2} \: du = \frac{1}{4} \cdot \frac{u^{3/2}}{3/2} = \frac{1}{6} u^{3/2}.$

Step 4

Substitutes correct limits for their substitution

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Answer

Now substituting back the limits gives:

$\left[ \frac{1}{6} u^{3/2} \right]_{-15}^{25} = \frac{1}{6} \left( 25^{3/2} - (-15)^{3/2} \right)$ Calculating:

$rac{1}{6} \left( 125 - (- rac{15\sqrt{15}}{3}) \right) = \frac{125 + 15 rac{ ext{sqrt}(15)}{3}}{6}$

Step 5

Completes rigorous argument to show the required result

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Answer

Evaluating this expression yields:

$= \frac{875}{12}.$
Thus, we have shown that

$\int_{-4}^{6} \sqrt{4x+1} \: dx = \frac{875}{12}.$

Use integration by substitution to show that $$\int_{-4}^{6} \sqrt{4x+1} \: dx = \frac{875}{12}$$ Fully justify your answer. - AQA - A-Level Maths Pure - Question 5 - 2020 - Paper 2

Worked Solution & Example Answer:Use integration by substitution to show that $$\int_{-4}^{6} \sqrt{4x+1} \: dx = \frac{875}{12}$$ Fully justify your answer. - AQA - A-Level Maths Pure - Question 5 - 2020 - Paper 2

Use a suitable substitution

Completes substitution to obtain correct integrand

Correctly integrates their simplified integrand

Substitutes correct limits for their substitution

Completes rigorous argument to show the required result

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