Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function:

Test 5 (Ch 5)
1. Express the limit as a definite integral on the given interval:

π‘™π‘–π‘š
π‘›β†’βˆž βˆ‘ π‘₯𝑖 𝑠𝑖𝑛 π‘₯𝑖 Ξ”π‘₯𝑛
𝑖=1 [0, πœ‹]

2. Express the integral as a limit of the Riemann sums. Do not evaluate the limit:
∫ π‘₯
1 + π‘₯5 𝑑π‘₯
8
1

3. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function:

𝑔(𝑦) = ∫ 𝑑2 𝑠𝑖𝑛 𝑑
𝑦
2
dt
4. Use Part 2 of the Fundamental Theorem of Calculus to evaluate the integral or explain why it does not exist:

∫ 𝑠𝑒𝑐2 𝑑
πœ‹/4
0
dt
5. Find the general indefinite integral:
∫(1 βˆ’ 3𝑑)(5 + 𝑑2)𝑑𝑑

6. Evaluate the integral:
∫ (10π‘₯ + 𝑒π‘₯)𝑑π‘₯
0
βˆ’1

7. Evaluate the indefinite integral:
∫ (𝑙𝑛 π‘₯)
π‘₯
3
𝑑π‘₯

8. Evaluate the indefinite integral:
∫ 𝑒π‘₯ √1 + 𝑒π‘₯𝑑π‘₯

9. Evaluate the definite integral, if it exists:

∫ (π‘₯ βˆ’ 1)9𝑑π‘₯
2
0

10. Find most general antiderivative of the function:

𝑓(𝑒) = 𝑒4+π‘’βˆšπ‘’
𝑒2