value of $A$.","markscheme":"- Substitutes limits 3 and 1 into their integrated expression from part (a): $2 \\ln(2(3)^3 + A) - 2 \\ln(2(1)^3 + A)$ M1\n- Simplifies to $2 \\ln(54 + A) - 2 \\ln(2 + A) = \\ln 196$ or $2 \\ln\\left(\\frac{54 + A}{2 + A}\\right) = \\ln 196$ M1\n- Uses power law for logs to obtain $\\ln\\left(\\frac{54 + A}{2 + A}\\right)^2 = \\ln 196$ and equates arguments: $\\left(\\frac{54 + A}{2 + A}\\right)^2 = 196$ M1\n- Takes the square root and solves the resulting linear equation $\\frac{54 + A}{2 + A} = 14 \\Rightarrow 54 + A = 28 + 14A \\Rightarrow 13A = 26$ M1\n- Finds $A = 2$ A1","marks":5,"order":1,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"analytical","label":"Analytical Integration and Logarithmic Laws","steps":[{"type":"text","order":0,"title":"Integrate the function","content":"To solve this problem, we first find the indefinite integral of the function:\n\n$$\\int \\frac{12t^2}{2t^3 + A} \\, dt$$\n\nWe notice that the numerator, $12t^2$, is a multiple of the derivative of the denominator:\n\n$$\\frac{d}{dt}(2t^3 + A) = 6t^2$$\n\nWe can apply integration by substitution. Let $u = 2t^3 + A$, which gives $du = 6t^2 \\, dt$, or $2 \\, du = 12t^2 \\, dt$. Substituting these into the integral gives:\n\n$$\\int \\frac{2}{u} \\, du = 2\\ln(2t^3 + A) + C$$\n\nSince $A > 0$ and $t \\ge 1$, the term $2t^3 + A$ is always positive, so we do not need absolute value bars."},{"type":"text","order":1,"title":"Apply limits of integration","content":"Now we substitute the limits of integration, $t = 3$ and $t = 1$, into our integrated expression:\n\n$$\\left[ 2\\ln(2t^3 + A) \\right]_{1}^{3} = 2\\ln(2(3)^3 + A) - 2\\ln(2(1)^3 + A)$$\n\nSimplifying the terms inside the logarithms:\n\n$$2\\ln(54 + A) - 2\\ln(2 + A)$$\n\nSince this definite integral is equal to $\\ln 196$, we can write:\n\n$$2\\ln(54 + A) - 2\\ln(2 + A) = \\ln 196$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=log-sub}{\\color{@sky}{2\\ln(54 + A) - 2\\ln(2 + A)}} = \\ln 196$"},{"latex":"$$\\htmlData{anchor=log-coeff}{\\color{@sky}{2}}\\ln\\left(\\frac{54 + A}{2 + A}\\right) = \\ln 196$"},{"latex":"$$\\ln\\left(\\frac{54 + A}{2 + A}\\right)^{\\htmlData{anchor=log-power}{\\color{@sky}{2}}} = \\ln 196$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"log-sub"},{"color":"sky","style":"text-color","anchor":"log-coeff"},{"color":"sky","style":"text-color","anchor":"log-power"}],"connections":[{"to":"log-coeff","from":"log-sub","color":"sky","label":"Factor out 2 and use quotient rule","style":"solid"},{"to":"log-power","from":"log-coeff","color":"sky","label":"Apply power law","style":"solid"}]},"order":2,"title":"Apply logarithmic laws","description":"We simplify the equation using the laws of logarithms."},{"type":"text","order":3,"title":"Solve the algebraic equation","content":"Since both sides of the equation are now in the form $\\ln(X) = \\ln(Y)$, we can equate their arguments:\n\n$$\\left(\\frac{54 + A}{2 + A}\\right)^2 = 196$$\n\nTaking the square root of both sides (since $A > 0$, the expression inside the parentheses is positive, so we take the positive square root of $196$, which is $14$):\n\n$$\\frac{54 + A}{2 + A} = 14$$\n\nNow, solve for $A$ by multiplying both sides by $(2 + A)$:\n\n$$54 + A = 14(2 + A)$$\n\n$$54 + A = 28 + 14A$$\n\n$$26 = 13A$$\n\n$$A = 2$$"},{"type":"text","order":4,"title":"State final answer","content":"The value of the positive constant $A$ is:\n\n$$A = 2$$"}]},{"id":"calculator","label":"Numerical Solver (GDC)","steps":[{"type":"text","order":0,"title":"Understand the equation","content":"We are given the following definite integral equation with a positive constant $A$:\n\n$$\\int_{1}^{3} \\frac{12t^2}{2t^3 + A} \\, dt = \\ln 196$$\n\nOur goal is to find the value of $A$ using the **Numerical Solver** on a Graphing Display Calculator (GDC). To do this, we can set up the equation directly on our GDC, replacing $A$ with $x$ as the unknown variable."},{"type":"text","order"

11.4 Reverse Chain Rule

Question 3

11.4 Reverse Chain Rule Questions

11.4 Reverse Chain Rule

Question 3

11.4 Reverse Chain Rule Questions