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
