Evaluate the indefinite integral: ∫ 𝑒𝑥 √1 + 𝑒𝑥𝑑𝑥

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]

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