x^2} dx$","markscheme":"$$u = 9 - x^2 \\implies \\frac{du}{dx} = -2x$ M1\nLimits: $x=2 \\implies u=5$; $x=0 \\implies u=9$ B1\n$\\int_9^5 \\frac{x}{u} \\cdot \\frac{1}{-2x} du = \\int_9^5 -\\frac{1}{2u} du$ M1\n$= \\int_5^9 \\frac{1}{2u} du$ M1\n$= [\\frac{1}{2}\\ln|u|]_5^9$ M1\n$= \\frac{1}{2}\\ln 9 - \\frac{1}{2}\\ln 5 = \\frac{1}{2}\\ln \\frac{9}{5}$ A2","marks":7,"order":0,"markingRubric":{"totalMarks":7,"points":[{"id":"p1","marks":1,"description":"Differentiate $u$ to find $du/dx = -2x$","code":"M"},{"id":"p2","marks":1,"description":"Change the limits correctly from $x=0, 2$ to $u=9, 5$","code":"B"},{"id":"p3","marks":1,"description":"Perform substitution to get the integral in $u$ only","code":"M"},{"id":"p4","marks":1,"description":"Address the negative constant and swap integration bounds","code":"M"},{"id":"p5","marks":1,"description":"Complete the integration to get a natural logarithm term","code":"M"},{"id":"p6","marks":2,"description":"Substitute the $u$ limits and simplify to the final answer","code":"A"}],"finalAnswers":["\\frac{1}{2}\\ln\\frac{9}{5}","\\ln\\sqrt{1.8}"],"generatedAt":"2026-05-06T14:04:39.325Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"limit_transformation","label":"Substitution with Limit Transformation","steps":[{"type":"text","order":0,"title":"Define the substitution","content":"We are given the substitution $u = 9 - x^2$. \n\nTo write the entire integral in terms of $u$, we need to find $du$ in terms of $dx$. Let's differentiate $u$ with respect to $x$:\n\n$$\\frac{du}{dx} = -2x$$\n\nRearranging this gives an expression for $dx$:\n\n$$dx = \\frac{du}{-2x}$$\n\nThis will allow us to substitute $dx$ and cancel out the $x$ variable in our integrand."},{"type":"text","order":1,"title":"Transform the limits","content":"Since we are doing a definite integration, we must transform the limits of integration from $x$ to $u$. \n\n* When the lower limit is $x = 0$:\n $$u = 9 - (0)^2 = 9$$\n\n* When the upper limit is $x = 2$:\n $$u = 9 - (2)^2 = 5$$\n\nSo, our new limits of integration in terms of $u$ are from $9$ to $5$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int_{\\htmlData{anchor=low-x}{0}}^{\\htmlData{anchor=up-x}{2}} \\frac{x}{\\htmlData{anchor=denom-x}{9 - x^2}} \\htmlData{anchor=dx-orig}{dx}$"},{"latex":"$$=\\int_{\\htmlData{anchor=low-u}{9}}^{\\htmlData{anchor=up-u}{5}} \\frac{x}{\\htmlData{anchor=denom-u}{u}} \\cdot \\left( \\htmlData{anchor=dx-sub}{\\frac{du}{-2x}} \\right)$"},{"latex":"$$=\\int_{9}^{5} -\\frac{1}{2u} du$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"low-u"},{"color":"amber","style":"text-color","anchor":"up-u"},{"color":"purple","style":"text-color","anchor":"denom-u"},{"color":"teal","style":"text-color","anchor":"dx-sub"}],"connections":[{"to":"low-u","from":"low-x","color":"sky","label":"x=0","style":"solid"},{"to":"up-u","from":"up-x","color":"amber","label":"x=2","style":"solid"},{"to":"denom-u","from":"denom-x","color":"purple","label":"u","style":"solid"},{"to":"dx-sub","from":"dx-orig","color":"teal","label":"dx","style":"solid"}]},"order":2,"title":"Substitute into the integral","description":"Now, we substitute $u$, $dx$, and our new limits of integration into the original integral. Notice how the $x$ terms nicely cancel out."},{"type":"text","order":3,"title":"Simplify the integral","content":"We can simplify the integral by reversing the limits. Remember that swapping the upper and lower limits changes the sign of the integral:\n\n$$\\int_a^b f(u) du = -\\int_b^a f(u) du$$\n\nApplying this rule to swap the limits from $[9, 5]$ to $[5, 9]$:\n\n$$\\int_9^5 -\\frac{1}{2u} du = \\int_5^9 \\frac{1}{2u} du$$\n\nThis makes the integration straightforward as we are integrating a positive term with standard limits."},{"type":"text","order":4,"title":"Integrate and evaluate","content":"Now, we integrate with respect to $u$. Recall that the antiderivative of $\\frac{1}{u}$ is $\\ln|u|$:\n\n$$\\int_5^9 \\frac{1}

Integration

Question 76

Integration Questions

Integration

Question 76

Integration Questions