math_三角升幂/降幂/微积分公式填空

三角函数

倍角

1. $sin2x=$
2. $cos2x=$

1. ​​2f-1​​
2. ​​1-2g​​

3. $tan2x=$

降幂(升角)公式

1. $sin^2x=$
2. $cos^2x=$
3. $2sin^2\frac{x}{2}=$
4. $2cos^2{\frac{x}{2}}=$
5. $1-cosx=$
6. $sinxcosx=$
7. $sin\frac{x}{2}cos\frac{x}{2}=$

积分

1. $\int kdx=$
2. $f \frac{1}{x}dx=$
3. $\int sinxdx=$
4. $\int cosxdx=$
5. $\int x^a dx=$
6. $\int a^x=$
7. $\int e^xdx=$
8. $\int \frac{1}{cos^2x}dx=\int sec^2xdx=$
9. $\int \frac{1}{sin^2}=\int csc^2xdx=$
10. $\int secxtanxdx=$
11. $\int cscxcotxdx=$
12. $\int \frac{1}{\sqrt{1-x^2}}dx=$
13. $\int \frac{1}{1+x^2}dx=$
14. $\int tanxdx=$
15. $\int cotxdx=$
16. $\int cscxdx=$
17. $\int secxdx=$
18. $\int \frac{1}{a^2+x^2}dx=$
19. $\int \frac{1}{x^2-a^2}dx=$
20. $\int \frac{1}{\sqrt{a^2-x^2}}dx=$
21. $\int \frac{1}{\sqrt{x^2+a^2}}dx=$
22. $\int \frac{1}{x^2-a^2}dx=$

参考答案

积分

1. $\int kdx=kx+C$
2. $\int dx=\int 1dx=x+C$
3. $\int 0dx=C$
4. $f \frac{1}{x}dx=ln|x|+C$
5. $\int sinxdx=-cosx+C$
6. $\int cosxdx=sinx+C$
7. $\int x^a dx=\frac{1}{a+1}x^{a+1}$
8. $\int a^x=\frac{a^x}{\ln a}+C(a>0;a\ne1)$
9. $\int e^xdx=e^x+C$
10. $\int \frac{1}{cos^2x}dx=\int sec^2xdx=tanx+C$
11. $\int \frac{1}{sin^2}=\int csc^2xdx=-cotx+C$
12. $\int secxtanxdx=secx+C$
13. $\int cscxcotxdx=-cscx+C$
14. $\int \frac{1}{\sqrt{1-x^2}}dx=arcsinx+C$
15. $\int \frac{1}{1+x^2}dx=arctanx+C$
16. $\int tanxdx=\int \frac{sinx}{cosx}dx=\int \frac{d(-cosx)}{sinx}=-\ln|cosx|+C$
17. $\int cotxdx=\ln |sinx|+C$
18. $\int cscxdx=\ln |cscx-cotx|+C$

1. $可由三角降角升幂和配凑乘以1=\frac{cosx}{cosx}$

19. $\int secxdx=\ln|secx+tanx|+C$
20. $\int \frac{1}{a^2+x^2}dx=\frac{1}{a}arctan{\frac{x}{a}}+C$
21. $\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}\ln|\frac{x-a}{x+a}|+C$

1. $可由\frac{1}{x^2-a^2}列项后分项积分得到$

22. $\int \frac{1}{\sqrt{a^2-x^2}}dx=arcsin(\frac{x}{a})+C$
23. $\int \frac{1}{\sqrt{x^2\pm a^2}}dx=\ln|x+\sqrt{x^2\pm a^2}|+C$

1. $p=\sqrt{x^2\pm a^2}$
2. $\int \frac{1}{\sqrt{x^2-a^2}}dx=\int \frac{1}{p}dx=\ln|x+p|+C$
3. 可由三角换元推导

24. $\int \sqrt{a^2-x^2}dx=\frac{a^2}{2}arcsin{\frac{x}{a}}+\frac{1}{2}\sqrt{a^2-x^2}+C=\frac{1}{2}(a^2arcsin{\frac{x}{a}}+\sqrt{a^2-x^2})+C$

1. $p=\sqrt{a^2-x^2}$
2. $\int pdx=\frac{1}{2}a^2\int \frac{1}{p}dx+\frac{1}{2}p+C=\frac{1}{2}(a^2\int \frac{1}{p}dx+p)+C$
3. 分部积分推导法

1. $\\ \begin{aligned} S&=\int \sqrt{x^2-a^2}dx \\ &=x\sqrt{x^2-a^2}-\int xd\sqrt{x^2-a^2} \end{aligned} \\为例方便说明推导和简洁性,提前给出如下标记(表达式记号) \\ \begin{aligned} \\A&=x\sqrt{x^2-a^2} \\B&=\int xd\sqrt{x^2-a^2} \\Q&=a^2\int\frac{1}{\sqrt{x^2-a^2}}dx=a^2\ln |x+\sqrt{x^2-a^2}| \end{aligned}$
   $\begin{aligned} B &=\int xd\sqrt{x^2-a^2}\\ &=\int \frac{x^2}{\sqrt{x^2-a^2}}dx \\ &\xlongequal{分子+0=-a^2+a^2}\int \frac{x^2-a^2+a^2}{\sqrt{x^2-a^2}}dx\\ &=\int\sqrt{x^2-a^2}dx+a^2\int\frac{1}{\sqrt{x^2-a^2}}dx\\ \\ &=S+Q \end{aligned}$
   $\\S=A-B=A-S-Q \\2S=A-Q \to S=\frac{1}{2}(A-Q) \\S=\frac{1}{2}(x\sqrt{x^2-a^2}-a^2\ln |x+\sqrt{x^2-a^2}|)$

25. $\int \sqrt{a^2+x^2}dx=\frac{1}{2}(x\sqrt{a^2+x^2}+a^2 \ln|\sqrt{a^2+x^2}+x|) +C$

1. 利用三角换元配合分部积分法可以推导
2. $\int sec^3t\ dt=\frac{1}{2}(secxtanx+\ln |secx+tanx|)+C$
3. $对于S=\int \sqrt{a^2\pm x^2}dx$

1. $p=\sqrt{a^2\pm x^2}$
2. $A=xp$
3. $Q=a^2\ln|x+p|$
4. $S=\frac{1}{2}(A\pm Q)+C$