Integrals

{"questions":[{"content":"Compute the definite integral $\\int_{0}^{1} 3x^2 \\, dx$.<br>[[input-1]]","widgets":{"input-1":{"type":"input","answer":"1"}},"explanation":"By evaluating the definite integral: $\\int_{0}^{1} 3x^2 \\, dx = \\left[ x^3 \\right]_{0}^{1} = 1^3 — 0^3 = 1$.","hints":["Use the power rule for integration.","Evaluate the integral from 0 to 1."]},{"content":"Compute the definite integral $\\int_{0}^{2} x \\, dx$.<br>[[input-2]]","widgets":{"input-2":{"type":"input","answer":"2"}},"explanation":"By evaluating the definite integral: $\\int_{0}^{2} x \\, dx = \\left[ \\frac{1}{2}x^2 \\right]_{0}^{2} = \\frac{1}{2}(2^2) — 0 = 2$.","hints":["Integrate $x$ and evaluate from 0 to 2.","Apply the power rule."]},{"content":"Compute the definite integral $\\int_{0}^{3} (2x + 1) \\, dx$.<br>[[input-3]]","widgets":{"input-3":{"type":"input","answer":"12"}},"explanation":"By evaluating the definite integral: $\\int_{0}^{3} (2x + 1) \\, dx = \\left[ x^2 + x \\right]_{0}^{3} = (3^2 + 3) — (0^2 + 0) = 12$.","hints":["Integrate each term separately and evaluate.","Use the power rule."]},{"content":"Compute the definite integral $\\int_{1}^{2} 4x^3 \\, dx$.<br>[[input-4]]","widgets":{"input-4":{"type":"input","answer":"15"}},"explanation":"By evaluating the definite integral: $\\int_{1}^{2} 4x^3 \\, dx = \\left[ x^4 \\right]_{1}^{2} = 2^4 — 1^4 = 16 — 1 = 15$.","hints":["Use the power rule for integration.","Evaluate the integral from 1 to 2."]},{"content":"Compute the definite integral $\\int_{-1}^{1} 6x \\, dx$.<br>[[input-5]]","widgets":{"input-5":{"type":"input","answer":"0"}},"explanation":"By evaluating the definite integral: $\\int_{-1}^{1} 6x \\, dx = \\left[ 3x^2 \\right]_{-1}^{1} = 3(1^2) — 3((-1)^2) = 3 — 3 = 0$.","hints":["Recognize the symmetry property of definite integrals.","Integrate $6x$ and evaluate between -1 and 1."]},{"content":"Compute the definite integral $\\int_{0}^{\\pi} \\sin x \\, dx$.<br>[[input-6]]","widgets":{"input-6":{"type":"input","answer":"2"}},"explanation":"By evaluating the definite integral: $\\int_{0}^{\\pi} \\sin x \\, dx = \\left[ -\\cos x \\right]_{0}^{\\pi} = (-\\cos \\pi) — (-\\cos 0) = 1 + 1 = 2$.","hints":["Integrate $\\sin x$ to get $-\\cos x$.","Evaluate between 0 and $\\pi$."]},{"content":"Compute the definite integral $\\int_{0}^{1} e^x \\, dx$.<br>[[input-7]]","widgets":{"input-7":{"type":"input","answer":"e — 1"}},"explanation":"By evaluating the definite integral: $\\int_{0}^{1} e^x \\, dx = \\left[ e^x \\right]_{0}^{1} = e^1 — e^0 = e — 1$.","hints":["The antiderivative of $e^x$ is itself.","Evaluate from 0 to 1."]},{"content":"Compute the definite integral $\\int_{1}^{2} \\frac{1}{x} \\, dx$.<br>[[input-8]]","widgets":{"input-8":{"type":"input","answer":"0.693"}},"explanation":"By evaluating the definite integral: $\\int_{1}^{2} \\frac{1}{x} \\, dx = \\left[ \\ln|x| \\right]_{1}^{2} = \\ln 2 — \\ln 1 = \\ln 2 \\approx 0.693$.","hints":["The antiderivative of $\\frac{1}{x}$ is the natural logarithm.","Evaluate from 1 to 2."]},{"content":"Compute the definite integral $\\int_{0}^{\\frac{\\pi}{2}} \\cos x \\, dx$.<br>[[input-9]]","widgets":{"input-9":{"type":"input","answer":"1"}},"explanation":"By evaluating the definite integral: $\\int_{0}^{\\frac{\\pi}{2}} \\cos x \\, dx = \\left[ \\sin x \\right]_{0}^{\\frac{\\pi}{2}} = \\sin \\frac{\\pi}{2} — \\sin 0 = 1 — 0 = 1$.","hints":["The antiderivative of $\\cos x$ is $\\sin x$.","Evaluate from 0 to $\\frac{\\pi}{2}$."]},{"content":"Compute the definite integral $\\int_{2}^{3} (x^2 + 3x + 2) \\, dx$.<br>[[input-10]]","widgets":{"input-10":{"type":"input","answer":"25"}},"explanation":"By evaluating the definite integral: $\\int_{2}^{3} (x^2 + 3x + 2) \\, dx = \\left[ \\frac{1}{3}x^3 + \\frac{3}{2}x^2 + 2x \\right]_{2}^{3} = 43\\frac{1}{3} — 18\\frac{1}{3} = 25$.","hints":["Integrate each term separately.","Use the power rule and evaluate from 2 to 3."]}],"mix":1}