Given f(x) = (1)/(x2) + 2x + 3, find the area under the graph of f over the interval [2, 5].

Step 1: Identify the function and the interval. The given function is f(x) = (1)/(5)x^2 + 3. The interval is [2, 5], so the lower limit is a=2 and the upper limit is b=5. Step 2: Set up the definite integral to find the area. The area under the graph of f(x) from a to b is given by the definite integral: Area = _a^b f(x)\,dx Substituting the given function and limits: Area = _2^5 ((1)/(5)x^2 + 3)\,dx Step 3: Find the antiderivative of the function. We integrate term by term: ((1)/(5)x^2 + 3)\,dx = (1)/(5) x^2\,dx + 3\,dx Using the power rule x^n\,dx = x^n+1n+1 and c\,dx = cx: (1)/(5) (x^2+12+1) + 3x = (1)/(5) ((x^3)/(3)) + 3x = (x^3)/(15) + 3x Let F(x) = (x^3)/(15) + 3x be the antiderivative. Step 4: Evaluate the antiderivative at the upper and lower limits. According to the Fundamental Theorem of Calculus, the definite integral is F(b) - F(a). Evaluate F(5): F(5) = (5^3)/(15) + 3(5) = (125)/(15) + 15 Simplify (125)/(15) by dividing both by 5: (25)/(3). F(5) = (25)/(3) + 15 Evaluate F(2): F(2) = (2^3)/(15) + 3(2) = (8)/(15) + 6 Step 5: Calculate the final result. Area = F(5) - F(2) = ((25)/(3) + 15) - ((8)/(15) + 6) Area = (25)/(3) + 15 - (8)/(15) - 6 Combine the constant terms and the fractional terms: Area = (15 - 6) + ((25)/(3) - (8)/(15)) Area = 9 + ((25 × 5)/(3 × 5) - (8)/(15)) Area = 9 + ((125)/(15) - (8)/(15)) Area = 9 + (117)/(15) Simplify (117)/(15) by dividing both by 3: (39)/(5). Area = 9 + (39)/(5) Convert 9 to a fraction with denominator 5: 9 = (45)/(5). Area = (45)/(5) + (39)/(5) = (45+39)/(5) = (84)/(5) The area under the graph is (84)/(5). Send me the next one 📸