Question

∫⁵ (2x²+3x+2) dx =?
²
Please solve this question.​

Answer 1

Answer:

$\boxed{\mathfrak{\int\limits^5_2 ({2x^2+3x+2}) \, dx = 115.5}}$

Explanation:

$\sf Compute \:  the \:  de finite \:  integral: \\  \sf \implies \int\limits^5_2 ({2x^2+3x+2}) \, dx  \\  \\  \sf Integrate  \: the  \: sum  \: term \:  by \:  term  \: and \\  \sf factor \:  out   \:  constants: \\  \sf \implies 2 \int\limits^5_2  {x}^{2}  \, dx + 3 \int\limits^5_2  x  \, dx + 2 \int\limits^5_2  1  \, dx$

$\sf Apply \: the \: fundamental \: theorem \: of \: calculus. \\ \sf The \: antiderivative \: of \: {x}^{2} \: is \: \frac{ {x}^{3} }{3} : \\ \\ \sf \implies \frac{2 {x}^{3} }{3} \Big|_2^5 + 3 \int\limits^5_2 x \, dx + 2 \int\limits^5_2 1 \, dx \\ \\ \sf Evaluate \: the \: antiderivative \: at \: the \: limits \: and \\ \sf subtract. \\ \sf \frac{2 {x}^{3} }{3} \Big|_2^5 = \frac{2 \times {5}^{3} }{3} - \frac{2 \times {2}^{3} }{3} = 78 : \\ \\ \sf \implies 78 + 3 \int\limits^5_2 x \, dx + 2 \int\limits^5_2 1 \, dx$

$\sf Apply \: the \: fundamental \: theorem \: of \: calculus. \\ \sf The \: antiderivative \: of \: x \: is \: \frac{ {x}^{2} }{2} : \\ \\ \sf \implies 78 + \frac{3 {x}^{2} }{2} \Big|_2^5 + 2 \int\limits^5_2 1 \, dx \\ \\ \sf Evaluate \: the \: antiderivative \: at \: the \: limits \: and \\ \sf subtract. \\ \sf \frac{3 {x}^{2} }{2} \Big|_2^5 = \frac{3 \times {5}^{2} }{2} - \frac{3 \times {2}^{2} }{2} = \frac{63}{2} : \\ \\ \sf \implies 78 + \frac{63}{2} + 2 \int\limits^5_2 1 \, dx$

$\sf Apply \: the \: fundamental \: theorem \: of \: calculus. \\ \sf The \: antiderivative \: of \: 1 \: is \: x : \\ \\ \sf \implies 78 + \frac{63}{2} + 2x \Big|_2^5 \\ \\ \sf Evaluate \: the \: antiderivative \: at \: the \: limits \: and \\ \sf subtract. \\ \sf 2x \Big|_2^5 = 2 \times 5 - 2 \times 2 = 6 : \\ \\ \sf \implies 78 + \frac{63}{2} + 6 \\ \\ \sf \implies \frac{156 + 63 + 12}{2} \\ \\ \sf \implies \frac{231}{2} \\ \\ \sf \implies 115.5$

Answer 2

Answer:

∫⁵ (2x²+3x+2) dx = 2x³/3 + 3x²/2 + 2x (from 2 to 5)

²

= [(2×5³)/3 + (3×5²)/2 + 2×5] - [(2×2³)/3 + (3×2²)/2 + 2×2]

By solving we get:

= 231/2