Definite integral T ^ (n-1) * f (x ^ N-t ^ n) upper limit x lower limit 0

According to the integral mean value theorem
The derivative is x ^ (n-1) f (0)

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1. If f (x) has continuous derivatives in [1, + ∞), and satisfies X-1 + X ∫ (upper limit x, lower limit 1) f (T) DT = (x + 1) ∫ (upper limit x, lower limit 1) TF (T) DT, find f (x) The answer is f (x) = x ^ (- 3) * e ^ (1-1 / x),
2. In the derivation process of [∫ (0, x) TF (T) DT] 'explained by experts, the result should be XF (x), or XF (x) -∫ (0, x) f (T) DT
3. Find: definite integral ∫ ((0, t) e ^ - x ^ 2 DX) (T
4. Find: definite integral ∫ (0 above, T below) (e ^ - x ^ 2) (t above, T below) DX
5. Calculation: definite integral ∫ (in the upper 1, in the lower √ 2 / 2) (√ 1-x ^ 2) / x ^ 2 DX to find the detailed process answer, please God
6. ∫ (3 sin T + 1 / 2 sin ^ 2 T) DT
7. D / dt ∫ sin (T ^ 2) DT (0 to 1),
8. To find the symmetry center of G (x) = 2 + X + sin (x + 1) requires specific steps and processes. Thank you!
9. Let f (x) be a differentiable function, definite integral (x, 0) (t-1) f (x-t) DT = 0, find f (x)
10. Finding the definite integral of (1 + T square) from x square to x cube under D / DX times ∫ DT / root sign
11. Finding the derivative of F (x) = sin ^ 3 · 1 / X
12. If f (x) is continuous and f (x) = x + (x ^ 2) ∫ (0,1) f (T) DT, find f (x)
13. The definite integral of the original function of the square of xsinx
14. It is proved that the continuous function f (x) satisfies: ∫ (0 to 1) f (TX) DT = f (x) + xsinx
15. Let f (x) satisfy ∫ f (TX) DT (from 0 to 1) = f (x) + xsinx
16. The expression of definite integral f (x) = ∫ 0 to 1| x-t | DT
17. ∫1/(1+cos t)dt
18. How to calculate the initial phase in simple harmonic motion? How to calculate the initial phase according to the image, we need to have the explanation of the unit circle, and we need to give examples The unit circle is mainly used to judge the positive and negative, the answer can not be too simple!
19. How is v = awcos (WT +?) derived from the expression of X in simple harmonic motion?
20. What's the relationship between COS and sin in the same corner? It's better to say how to prove it Better be specific,