frac{1}{4} \\int \\frac{1}{u} \\, du = \\frac{1}{4}\\ln|u| + C$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{1}{4}\\ln|\\htmlData{anchor=final-u}{\\color{@sky}{u}}| + C$"},{"text":"Replacing $u$ with our original definition $u = 4x - 5$:"},{"latex":"$$\\frac{1}{4}\\ln|\\htmlData{anchor=final-ans}{\\color{@sky}{4x - 5}}| + C$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"final-u"},{"color":"sky","style":"text-color","anchor":"final-ans"}],"connections":[{"to":"final-ans","from":"final-u","color":"sky","label":"substitute back","style":"solid"}]},"order":4,"title":"Substitute back to x","description":"Finally, we substitute our original expression in $x$ back into the result."}]},{"id":"4345a58e-3a7d-43ef-85bd-eec65a69839f:direct_integration","label":"(b) Direct Integration","steps":[{"type":"text","order":0,"title":"Identify the standard rule","content":"We want to find the integral of the given expression:\n\n$$\\int \\frac{1}{4x - 5} \\, dx$$\n\nThis expression has a linear denominator of the form $ax + b$. We can directly integrate this using the standard logarithmic integration formula:\n\n$$\\int \\frac{1}{ax+b} \\, dx = \\frac{1}{a}\\ln|ax+b| + C$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\int \\frac{1}{\\htmlData{anchor=a-formula}{\\color{@sky}{a}}x + b} \\, dx = \\frac{1}{\\htmlData{anchor=a-coeff}{\\color{@sky}{a}}}\\ln|ax+b| + C$$"},{"latex":"$$$\\text{In our expression, the coefficient of } x \\text{ is } \\htmlData{anchor=a-val}{\\color{@sky}{4}}$$"},{"latex":"$$$\\int \\frac{1}{\\color{@sky}{4}x - 5} \\, dx = \\frac{1}{\\htmlData{anchor=a-result}{\\color{@sky}{4}}}\\ln|4x-5| + C$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-formula"},{"color":"sky","style":"text-color","anchor":"a-coeff"},{"color":"sky","style":"text-color","anchor":"a-val"},{"color":"sky","style":"text-color","anchor":"a-result"}],"connections":[{"to":"a-val","from":"a-formula","color":"sky","label":"identify a","style":"solid"},{"to":"a-result","from":"a-val","color":"sky","label":"substitute","style":"solid"}]},"order":1,"title":"Apply direct integration","description":"By matching the denominator $4x - 5$ to $ax + b$, we identify that $a = 4$. Substituting this into our integration formula yields the integrated logarithm term with its correct scale factor."},{"type":"text","order":2,"title":"State final answer and marks","content":"Let's review how marks are awarded for this problem to ensure we have met all the criteria:\n\n* **M1**: Integrating the expression to a natural logarithm form: $\\ln|4x-5|$\n* **A1**: Having the correct coefficient of $\\frac{1}{4}$ in front of the logarithm\n* **B1**: Writing down the constant of integration, $+ C$\n\nPutting everything together, we get our final answer:\n\n$$\\frac{1}{4}\\ln|4x-5| + C$$"}]}],"generatedAt":"2026-06-09T05:58:45.677Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":86}},{"id":"7242d354-9652-4ce4-b93c-a86a27c4f362","revisionId":"b957f197-4c86-4b3b-96d2-9f36989e55e0","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"easy","totalMarks":5,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"779b6cfe-e921-4545-a5d0-674afa30e173","label":"a","content":"Find: $\\int_0^2 \\frac{12}{(3x + 2)^2} dx$","markscheme":"- Rewrites as $12(3x+2)^{-2}$ and integrates: $\\left[ 12(3x+2)^{-1} \\times \\frac{1}{3} \\times (-1) \\right]_0^2 = \\left[ -4(3x+2)^{-1} \\right]_0^2$ M1 A1\n- Substitutes limits: $\\left[ -4(8)^{-1} \\right] - \\left[ -4(2)^{-1} \\right]$ M1\n- Calculates values: $-\\frac{1}{2} + 2$ M1\n- Final answer $\\frac{3}{2}$ or $1.5$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Evidence of rewriting the integrand as a power and attempting to integrate to the form $k(3x+2)^{-1}$.","code":"M"},{"id":"p2","marks":1,"description":"Correct integration including the factor of 1/3, resulting in $-4

1.8 Integration

Question 4

1.8 Integration Questions

1.8 Integration

Question 4

1.8 Integration Questions