Some students took the power train to travel. When the train accelerated evenly on a straight track, one student proposed, "can we use the equipment around us to measure the acceleration of the train?" many students participated in the measurement. The measurement process was as follows: they looked out of the window every 100m   While recording the time with a watch, they observed that the time interval from the first landmark to the second landmark is 5S, and the time interval from the first landmark to the third landmark is 9s. Please find out according to their measurement: (1) Acceleration of train; (2) The speed at which they reached the third sign

Some students took the power train to travel. When the train accelerated evenly on a straight track, one student proposed, "can we use the equipment around us to measure the acceleration of the train?" many students participated in the measurement. The measurement process was as follows: they looked out of the window every 100m   While recording the time with a watch, they observed that the time interval from the first landmark to the second landmark is 5S, and the time interval from the first landmark to the third landmark is 9s. Please find out according to their measurement: (1) Acceleration of train; (2) The speed at which they reached the third sign

(1) Study the process of the train from passing the first road sign to passing the second road sign, with displacement X1 = 100m and time T1 = 5S,
Then the speed at the middle time is V1 = x1
t1
Similarly, study the process of the train from passing the second road sign to passing the third road sign, with displacement x2 = 100m and time T2 = 4S,
Then the speed at the middle time is V2 = x2
t2
Then the acceleration of the train is: a = V2 − v1
t1
2+t2
two
Substitute into the solution: a = 1.11m/s2
(2) According to the kinematic formula, the speed of the train passing the third road sign V3 = V2 + at2
2=27.2m/s
A: (1) the acceleration of the train is 1.11m/s2;
(2) When they reached the third road sign, the speed was 27.2m/s

The indefinite integral of X / (radical 5-x) is evaluated by partial integral and second transformation method respectively

∫xdx/√(5-x)
=-2∫xd√(5-x)
=-2x√(5-x) + 2∫√(5-x) dx
=-2x√(5-x) - (4/3).[(5-x)]^(3/2) + C

Solving indefinite integral ∫ DX / x + root sign (x ^ 2 + 1) by substitution method

Let x = tan α Then √ (x) ²+ 1）=1/cos α
‡ original formula = ∫ D (tan) α）/ （tan α+ 1/cos α）
=∫（1/cos ²α）/ （tan α+ 1/cos α） d α
=∫（cos α） d α/ （sin α cos ²α+ cos ²α）
=∫d（sin α）/ 【sin α （1-sin ²α）+ 1-sin ²α】
=-1/【2（sin α+ 1）】-1/4ln〡（sin α- 1）/（sin α+ 1）〡+C
Due to sin α= x/（√（x ²+ 1) ) so
Original formula = - 1 / [2 (x / √ (x ²+ 1））+2】-1/4ln〡（x/（√（x ²+ 1））-1）/（x/（√（x ²+ 1））+1）〡
+C
Finally done!

(1-x ^ 2 under the root sign) / X for indefinite integral

Replace x with SiNx or cosx

Find the indefinite integral of (1-x ^ 2) / x ^ 2 under the root sign The denominator x ^ 2 is not in the root sign~

Let x = sin α, Then DX = cos α* d α
∫√(1-x^2) *dx /x^2
=∫cos α * (cos α * d α) / (sin α)^ two
=∫(cot α)^ 2 d α
=∫[(csc α)^ 2 -1] *d α
=∫(csc α)^ 2*d α - ∫ d α
=-cot α - α + C
=-√(1-x^2)/x - arcsin α + C

How to calculate indefinite integral with 1 + x ^ 2 under x ^ 2) / radical Specific steps are required

Another root sign (1 + x) ²）= T original formula = ∫ root sign (T) ²- 1) DT another root sign (T) ²- 1）=tan β t=sec β Then the original formula = ∫ 1 / cos ³β d β = ∫tan β sec β + sec β d β = sec β + ln|sec β+ tan β| + C = t + ln|t + root sign (T) ²- 1）| + c...