Calculus

10 calculus questions

Test 5 (Ch 5)
1. Express the limit as a definite integral on the given interval:
𝑙𝑖𝑚
𝑛→∞ ∑ 𝑥𝑖 𝑠𝑖𝑛 𝑥𝑖 Δ𝑥𝑛
𝑖=1 [0, 𝜋]

2. Express the integral as a limit of the Riemann sums. Do not evaluate the limit:
   ∫ 𝑥
   1 + 𝑥5 𝑑𝑥
   8
   1
3. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function:
   𝑔(𝑦) = ∫ 𝑡2 𝑠𝑖𝑛 𝑡
   𝑦
   2dt
4. Use Part 2 of the Fundamental Theorem of Calculus to evaluate the integral or explain why it does not exist:
   ∫ 𝑠𝑒𝑐2 𝑡
   𝜋/4
   0dt
5. Find the general indefinite integral: ∫(1 − 3𝑡)(5 + 𝑡2)𝑑𝑡
6. Evaluate the integral:
   ∫ (10𝑥 + 𝑒𝑥)𝑑𝑥
   0
   −1
7. Evaluate the indefinite integral:
   ∫ (𝑙𝑛 𝑥)
   𝑥
   3
   𝑑𝑥
8. Evaluate the indefinite integral: ∫ 𝑒𝑥 √1 + 𝑒𝑥𝑑𝑥
9. Evaluate the definite integral, if it exists:
   ∫ (𝑥 − 1)9𝑑𝑥
   2
   0
10. Find most general anti-derivative of the function:
    𝑓(𝑢) = 𝑢4+𝑢√𝑢
    𝑢2