• math_三角升幂/降幂/微积分公式填空

    三角函数

    倍角

    1. s i n 2 x = sin2x= sin2x=
    2. c o s 2 x = cos2x= cos2x=
      1. 2f-1
      2. 1-2g
    3. t a n 2 x = tan2x= tan2x=

    降幂(升角)公式

    1. s i n 2 x = sin^2x= sin2x=

    2. c o s 2 x = cos^2x= cos2x=

    3. 2 s i n 2 x 2 = 2sin^2\frac{x}{2}= 2sin22x​=

    4. 2 c o s 2 x 2 = 2cos^2{\frac{x}{2}}= 2cos22x​=

    5. 1 − c o s x = 1-cosx= 1−cosx=

    6. s i n x c o s x = sinxcosx= sinxcosx=

    7. s i n x 2 c o s x 2 = sin\frac{x}{2}cos\frac{x}{2}= sin2x​cos2x​=

    积分

    1. ∫ k d x = \int kdx= ∫kdx=
    2. f 1 x d x = f \frac{1}{x}dx= fx1​dx=
    3. ∫ s i n x d x = \int sinxdx= ∫sinxdx=
    4. ∫ c o s x d x = \int cosxdx= ∫cosxdx=
    5. ∫ x a d x = \int x^a dx= ∫xadx=
    6. ∫ a x = \int a^x= ∫ax=
    7. ∫ e x d x = \int e^xdx= ∫exdx=
    8. ∫ 1 c o s 2 x d x = ∫ s e c 2 x d x = \int \frac{1}{cos^2x}dx=\int sec^2xdx= ∫cos2x1​dx=∫sec2xdx=
    9. ∫ 1 s i n 2 = ∫ c s c 2 x d x = \int \frac{1}{sin^2}=\int csc^2xdx= ∫sin21​=∫csc2xdx=
    10. ∫ s e c x t a n x d x = \int secxtanxdx= ∫secxtanxdx=
    11. ∫ c s c x c o t x d x = \int cscxcotxdx= ∫cscxcotxdx=
    12. ∫ 1 1 − x 2 d x = \int \frac{1}{\sqrt{1-x^2}}dx= ∫1−x2 ​1​dx=
    13. ∫ 1 1 + x 2 d x = \int \frac{1}{1+x^2}dx= ∫1+x21​dx=
    14. ∫ t a n x d x = \int tanxdx= ∫tanxdx=
    15. ∫ c o t x d x = \int cotxdx= ∫cotxdx=
    16. ∫ c s c x d x = \int cscxdx= ∫cscxdx=
    17. ∫ s e c x d x = \int secxdx= ∫secxdx=
    18. ∫ 1 a 2 + x 2 d x = \int \frac{1}{a^2+x^2}dx= ∫a2+x21​dx=
    19. ∫ 1 x 2 − a 2 d x = \int \frac{1}{x^2-a^2}dx= ∫x2−a21​dx=
    20. ∫ 1 a 2 − x 2 d x = \int \frac{1}{\sqrt{a^2-x^2}}dx= ∫a2−x2 ​1​dx=
    21. ∫ 1 x 2 + a 2 d x = \int \frac{1}{\sqrt{x^2+a^2}}dx= ∫x2+a2 ​1​dx=
    22. ∫ 1 x 2 − a 2 d x = \int \frac{1}{x^2-a^2}dx= ∫x2−a21​dx=

    参考答案

    math_高数公式每日一过_part2(private)_xuchaoxin1375的博客-CSDN博客

    积分

    1. ∫ k d x = k x + C \int kdx=kx+C ∫kdx=kx+C

    2. ∫ d x = ∫ 1 d x = x + C \int dx=\int 1dx=x+C ∫dx=∫1dx=x+C

    3. ∫ 0 d x = C \int 0dx=C ∫0dx=C

    4. f 1 x d x = l n ∣ x ∣ + C f \frac{1}{x}dx=ln|x|+C fx1​dx=ln∣x∣+C

    5. ∫ s i n x d x = − c o s x + C \int sinxdx=-cosx+C ∫sinxdx=−cosx+C

    6. ∫ c o s x d x = s i n x + C \int cosxdx=sinx+C ∫cosxdx=sinx+C

    7. ∫ x a d x = 1 a + 1 x a + 1 \int x^a dx=\frac{1}{a+1}x^{a+1} ∫xadx=a+11​xa+1

    8. ∫ a x = a x ln ⁡ a + C ( a > 0 ; a ≠ 1 ) \int a^x=\frac{a^x}{\ln a}+C(a>0;a\ne1) ∫ax=lnaax​+C(a>0;a=1)

    9. ∫ e x d x = e x + C \int e^xdx=e^x+C ∫exdx=ex+C

    10. ∫ 1 c o s 2 x d x = ∫ s e c 2 x d x = t a n x + C \int \frac{1}{cos^2x}dx=\int sec^2xdx=tanx+C ∫cos2x1​dx=∫sec2xdx=tanx+C

    11. ∫ 1 s i n 2 = ∫ c s c 2 x d x = − c o t x + C \int \frac{1}{sin^2}=\int csc^2xdx=-cotx+C ∫sin21​=∫csc2xdx=−cotx+C

    12. ∫ s e c x t a n x d x = s e c x + C \int secxtanxdx=secx+C ∫secxtanxdx=secx+C

    13. ∫ c s c x c o t x d x = − c s c x + C \int cscxcotxdx=-cscx+C ∫cscxcotxdx=−cscx+C

    14. ∫ 1 1 − x 2 d x = a r c s i n x + C \int \frac{1}{\sqrt{1-x^2}}dx=arcsinx+C ∫1−x2 ​1​dx=arcsinx+C

    15. ∫ 1 1 + x 2 d x = a r c t a n x + C \int \frac{1}{1+x^2}dx=arctanx+C ∫1+x21​dx=arctanx+C

    16. ∫ t a n x d x = ∫ s i n x c o s x d x = ∫ d ( − c o s x ) s i n x = − ln ⁡ ∣ c o s x ∣ + C \int tanxdx=\int \frac{sinx}{cosx}dx=\int \frac{d(-cosx)}{sinx}=-\ln|cosx|+C ∫tanxdx=∫cosxsinx​dx=∫sinxd(−cosx)​=−ln∣cosx∣+C

    17. ∫ c o t x d x = ln ⁡ ∣ s i n x ∣ + C \int cotxdx=\ln |sinx|+C ∫cotxdx=ln∣sinx∣+C

    18. ∫ c s c x d x = ln ⁡ ∣ c s c x − c o t x ∣ + C \int cscxdx=\ln |cscx-cotx|+C ∫cscxdx=ln∣cscx−cotx∣+C

      1. 可由三角降角升幂和配凑乘以 1 = c o s x c o s x 可由三角降角升幂和配凑乘以1=\frac{cosx}{cosx} 可由三角降角升幂和配凑乘以1=cosxcosx​

    19. ∫ s e c x d x = ln ⁡ ∣ s e c x + t a n x ∣ + C \int secxdx=\ln|secx+tanx|+C ∫secxdx=ln∣secx+tanx∣+C

    20. ∫ 1 a 2 + x 2 d x = 1 a a r c t a n x a + C \int \frac{1}{a^2+x^2}dx=\frac{1}{a}arctan{\frac{x}{a}}+C ∫a2+x21​dx=a1​arctanax​+C

    21. ∫ 1 x 2 − a 2 d x = 1 2 a ln ⁡ ∣ x − a x + a ∣ + C \int \frac{1}{x^2-a^2}dx=\frac{1}{2a}\ln|\frac{x-a}{x+a}|+C ∫x2−a21​dx=2a1​ln∣x+ax−a​∣+C

      1. 可由 1 x 2 − a 2 列项后分项积分得到 可由\frac{1}{x^2-a^2}列项后分项积分得到 可由x2−a21​列项后分项积分得到

    22. ∫ 1 a 2 − x 2 d x = a r c s i n ( x a ) + C \int \frac{1}{\sqrt{a^2-x^2}}dx=arcsin(\frac{x}{a})+C ∫a2−x2 ​1​dx=arcsin(ax​)+C

    23. ∫ 1 x 2 ± a 2 d x = ln ⁡ ∣ x + x 2 ± a 2 ∣ + C \int \frac{1}{\sqrt{x^2\pm a^2}}dx=\ln|x+\sqrt{x^2\pm a^2}|+C ∫x2±a2 ​1​dx=ln∣x+x2±a2 ​∣+C

      1. p = x 2 ± a 2 p=\sqrt{x^2\pm a^2} p=x2±a2 ​
      2. ∫ 1 x 2 − a 2 d x = ∫ 1 p d x = ln ⁡ ∣ x + p ∣ + C \int \frac{1}{\sqrt{x^2-a^2}}dx=\int \frac{1}{p}dx=\ln|x+p|+C ∫x2−a2 ​1​dx=∫p1​dx=ln∣x+p∣+C
      3. 可由三角换元推导

    24. ∫ a 2 − x 2 d x = a 2 2 a r c s i n x a + 1 2 a 2 − x 2 + C = 1 2 ( a 2 a r c s i n x a + a 2 − x 2 ) + C \int \sqrt{a^2-x^2}dx=\frac{a^2}{2}arcsin{\frac{x}{a}}+\frac{1}{2}\sqrt{a^2-x^2}+C=\frac{1}{2}(a^2arcsin{\frac{x}{a}}+\sqrt{a^2-x^2})+C ∫a2−x2 ​dx=2a2​arcsinax​+21​a2−x2 ​+C=21​(a2arcsinax​+a2−x2 ​)+C

      1. p = a 2 − x 2 p=\sqrt{a^2-x^2} p=a2−x2 ​

      2. ∫ p d x = 1 2 a 2 ∫ 1 p d x + 1 2 p + C = 1 2 ( a 2 ∫ 1 p d x + p ) + C \int pdx=\frac{1}{2}a^2\int \frac{1}{p}dx+\frac{1}{2}p+C=\frac{1}{2}(a^2\int \frac{1}{p}dx+p)+C ∫pdx=21​a2∫p1​dx+21​p+C=21​(a2∫p1​dx+p)+C

      3. 分部积分推导法

        1. S = ∫ x 2 − a 2 d x = x x 2 − a 2 − ∫ x d x 2 − a 2 为例方便说明推导和简洁性 , 提前给出如下标记 ( 表达式记号 ) A = x x 2 − a 2 B = ∫ x d x 2 − a 2 Q = a 2 ∫ 1 x 2 − a 2 d x = a 2 ln ⁡ ∣ x + x 2 − a 2 ∣ \\ S=∫√x2−a2dx=x√x2−a2−∫xd√x2−a2 \\为例方便说明推导和简洁性,提前给出如下标记(表达式记号) \\ A=x√x2−a2B=∫xd√x2−a2Q=a2∫1√x2−a2dx=a2ln|x+√x2−a2| S​=∫x2−a2 ​dx=xx2−a2 ​−∫xdx2−a2 ​​为例方便说明推导和简洁性,提前给出如下标记(表达式记号)ABQ​=xx2−a2 ​=∫xdx2−a2 ​=a2∫x2−a2 ​1​dx=a2ln∣x+x2−a2 ​∣​

          B = ∫ x d x 2 − a 2 = ∫ x 2 x 2 − a 2 d x = 分子 + 0 = − a 2 + a 2 ∫ x 2 − a 2 + a 2 x 2 − a 2 d x = ∫ x 2 − a 2 d x + a 2 ∫ 1 x 2 − a 2 d x = S + Q B=∫xd√x2−a2=∫x2√x2−a2dx\xlongequal分子+0=−a2+a2∫x2−a2+a2√x2−a2dx=∫√x2−a2dx+a2∫1√x2−a2dx=S+Q B​=∫xdx2−a2 ​=∫x2−a2 ​x2​dx分子+0=−a2+a2 ∫x2−a2 ​x2−a2+a2​dx=∫x2−a2 ​dx+a2∫x2−a2 ​1​dx=S+Q​

          S = A − B = A − S − Q 2 S = A − Q → S = 1 2 ( A − Q ) S = 1 2 ( x x 2 − a 2 − a 2 ln ⁡ ∣ x + x 2 − a 2 ∣ ) \\S=A-B=A-S-Q \\2S=A-Q \to S=\frac{1}{2}(A-Q) \\S=\frac{1}{2}(x\sqrt{x^2-a^2}-a^2\ln |x+\sqrt{x^2-a^2}|) S=A−B=A−S−Q2S=A−Q→S=21​(A−Q)S=21​(xx2−a2 ​−a2ln∣x+x2−a2 ​∣)

    25. ∫ a 2 + x 2 d x = 1 2 ( x a 2 + x 2 + a 2 ln ⁡ ∣ a 2 + x 2 + x ∣ ) + C \int \sqrt{a^2+x^2}dx=\frac{1}{2}(x\sqrt{a^2+x^2}+a^2 \ln|\sqrt{a^2+x^2}+x|) +C ∫a2+x2 ​dx=21​(xa2+x2 ​+a2ln∣a2+x2 ​+x∣)+C

      1. 利用三角换元配合分部积分法可以推导
      2. ∫ s e c 3 t   d t = 1 2 ( s e c x t a n x + ln ⁡ ∣ s e c x + t a n x ∣ ) + C \int sec^3t\ dt=\frac{1}{2}(secxtanx+\ln |secx+tanx|)+C ∫sec3t dt=21​(secxtanx+ln∣secx+tanx∣)+C
      3. 对于 S = ∫ a 2 ± x 2 d x 对于S=\int \sqrt{a^2\pm x^2}dx 对于S=∫a2±x2 ​dx
        1. p = a 2 ± x 2 p=\sqrt{a^2\pm x^2} p=a2±x2 ​
        2. A = x p A=xp A=xp
        3. Q = a 2 ln ⁡ ∣ x + p ∣ Q=a^2\ln|x+p| Q=a2ln∣x+p∣
        4. S = 1 2 ( A ± Q ) + C S=\frac{1}{2}(A\pm Q)+C S=21​(A±Q)+C
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  • 原文地址：https://blog.csdn.net/xuchaoxin1375/article/details/126546743