## hartnn 2 years ago $\qquad \qquad \qquad \qquad \qquad \huge{\color{Blue}{T}\color{Yellow}{U}\color{Magenta}{T}\color{Pink}{O}\color{MidnightBlue}{R} I\color{Blue}{A}\color{Green}{L}}$ $\text{Integration Formulae and Tips to solve certain types of integrals.} \\\text{Also look for tables giving standard substitutions.} \\ \text{*Spoiler alert*: Long tutorial.}$

1. hartnn

$$\huge \color{green}{\star \text{List of Integration Formulas}\star} \\ \large \boxed{ \frac{d}{\:dx}[f(x)]=g(x) \implies \int g(x)\:dx=f(x)+c \\ \text{where c is a constant} } \\~ \\~ \\~ \\~\\ \huge 1. \int x^n \:dx=\frac{x^{n+1}}{n+1}+c \\ example(a) : \int 1\:dx =\int x^0\:dx=\frac{x^1}{1}+c=x+c \\ example(b) : \int \frac{1}{\sqrt x}\:dx =\int x^{-\frac{1}{2}}\:dx=\frac{x^{\frac{1}{2}}}{\frac{1}{2}}+c=2\sqrt x+c \\~ \\~ \\~ \\ \huge 2. If \quad \int g(x)\:dx=f(x)+c \\ \huge then \quad \int g(ax+b)\:dx=\frac{f(ax+b)}{a}+c \\~ \\~ \\ \huge 3. (a) \: \int a^x \:dx=\frac{a^x}{\ln \: a}+c \\ \huge (b)\: \int e^x\:dx=e^x+c \\~ \\~ \\\huge 4. \: \int \frac{1}{x}\:dx=\ln |x|+c\\ \huge \color{red}{ \\ \\ \huge \text{In general,}\int \frac{f’(x)}{f(x)}\:dx=\ln|f(x)|+c } \\~ \\~ \\~ \\~\\~ \\~ \\ \huge \quad \quad \quad \quad \color{blue}{\text{Trigonometric Integrals}} \\ \huge 5. \int \sin\: x\:dx=-\cos\:x+c \\ \huge 6. \int \cos\: x\:dx=\sin\:x+c \\ \huge 7. \int \tan\: x\:dx=\ln|\sec\:x|+c \\ \huge 8. \int \cot\: x\:dx=\ln|\sin\:x|+c \\ \large 9. \int \sec\: x\:dx=\ln|\sec\:x+\tan\:x|+c=\ln|\tan(\frac{\pi}{4}+\frac{x}{2})|+c \\ \large 10. \int \csc\: x\:dx=\ln|\csc\:x+\cot\:x|+c=\ln|\tan(\frac{x}{2})|+c \\ \huge 11.\int \sec^2 x\:dx=\tan\:x+c \\ \huge 12.\int \csc^2 x\:dx=-\cot\:x+c \\ \huge 13.\int \sec\: x\:.\tan\:x \:dx=\sec\:x+c \\ \huge 14.\int \csc\: x.\:\cot\:x \:dx=-\csc\:x+c \\ \huge 15. \int \frac{1}{\sqrt{a^2-x^2}}dx=\sin^{-1}\frac{x}{a}+c \\ \huge 16. \int \frac{1}{x^2+a^2}dx=a^{-1}\tan^{-1}(\frac{x}{a})+c \\ \huge 17. \int \frac{1}{x\sqrt{x^2-a^2}}dx=a^{-1}\sec^{-1}(\frac{x}{a})+c \\~ \\~ \\~ \\~ \\ \huge 18. \int \frac{1}{\sqrt{x^2+a^2}}dx=\ln|x+\sqrt{x^2+a^2}|+c \\ \huge 19. \int \frac{1}{ {x^2-a^2}}dx=\ln|x+\sqrt{x^2-a^2}|+c \\ \huge 20. \int \frac{1}{ {x^2-a^2}}dx=\frac{1}{2a}\ln|\frac{x-a}{x+a}|+c \\ \huge 21. \int \frac{1}{ {a^2-x^2}}dx=\frac{1}{2a}\ln|\frac{x+a}{x-a}|+c \\ \large 22. \int \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln| x+\sqrt{x^2+a^2}|+c \\ \large 23. \int \sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\ln| x+\sqrt{x^2-a^2}|+c \\ \large 24. \int \sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\color{red}{\sin^{-1}\frac{x}{a}}+c \\~ \\~ \\~ \\~ \\~ \\~ \\~ \\~ \\ \huge 25. \quad \quad \quad \quad \color{blue}{ \text{Product Rule}} \\ \text{u and v are functions of x} \\ \huge \int uv\:dx=u\int v\:dx-\int(\frac{du}{dx}\int v.dx)dx \\ \huge \int (u.\frac{dv}{dx})\:dx=uv-\int (v.\frac{du}{dx})dx \\ \huge 26. \color{red}{\int e^x[f(x)+f’(x)]dx=e^xf(x)+c} \\ \huge 27. \int \ln \:x \: dx=x(ln\:x-1)+c \\~ \\~ \\~ \\~ \boxed{ \\ \large 28. \int e^{ax}\sin(bx)dx=e^{ax}\frac{a\sin(bx)-b\cos(bx)}{a^2+b^2}+c \\ \large 29. \int e^{ax}\cos(bx)dx=e^{ax}\frac{a\cos(bx)+b\sin(bx)}{a^2+b^2}+c } \\~ \\~ \\~ \\~ \\ \boxed { \\ \huge \quad \quad \quad \quad \color{blue}{ \text{Reduction Formula }} \\ \large 30. \int \cos^nx\:dx= \frac{\sin\: x.\: \cos^{n-1}x}{n}+\frac{n-1}{n}\int \cos^{n-2}x\:dx \\ \large 31. \int \sin^nx\:dx= \frac{-\cos\: x.\: \sin^{n-1}x}{n}+\frac{n-1}{n}\int \sin^{n-2}x\:dx }$$ |dw:1352010795406:dw| $$\\~ \\~ \\~ \\~ \large \color{green}{\star \text{Tips to solve certain types of Integrals}\star } \\ \large \color{blue}{\text{N=Numerator,D=Denominator}} \\ \text{1. To integrate} \huge \frac{1}{a\sin\:x+b\cos\:x+c}\\ \text{put, t=tan(x/2),then} \large \sin\:x =\frac{2t}{1+t^2} \quad \cos\:x=\frac{1-t^2}{1+t^2} \quad dx=\frac{2}{1+t^2} \\ \text{2. To integrate} \huge \frac{1}{a\sin\:x+b\cos\:x}\\ \text{Multiply and divide by }\large \sqrt{a^2+b^2}\text{in the D and express D as } \\ \large \sin(x\pm \alpha) or \cos(x\pm \alpha) \\ \text{3. To integrate} \huge \frac{1}{a\sin^2x+b\cos^2x}\\ \quad \text{divide N and D by} \large \cos^2x \text{then,put} \quad t=\tan \:x \\ \text{4. To integrate} \huge \frac{c\sin\:x+d\cos\:x}{a\sin\:x+b\cos\:x} or \frac{ce^x+d}{ae^x+b}\\ \text{express N as} \large A(D)+B\frac{d}{dx}(D) \\~ \\~ \\~ \\~ \\ \text{5. To integrate even powers of sine and cosine, use} \\ \huge sin^2x=\frac{1-cos2x}{2},\quad cos^2x=\frac{1+cos2x}{2} \\~ \\ \text{6. To integrate odd powers of sine and cosine,}\\ \text{ split the odd power into even power and unit power and put t=co-function} \\ \huge cos^5x=cos\:x.cos^4x,t=sin\:x \\~ \\ \text{7. To integrate any power of tan x(or cot x), }\\ \text{(i)Separate out }\quad \huge \tan^2x \\ \text{(ii)Write it as} \huge \sec^2x-1; \\ \text{(iii)Split it in 2 integrals} \\ \text{(iv)put t=tan x in integrals where } \huge \sec^2x\:dx \quad \text{is present} \\~ \\ \text{8. To integrate odd power of sec x(or csc x), }\\ \text{(i)Separate out }\quad \huge \sec^2x \\ \text{(ii)Write it as} \huge 1+\tan^2x; \\ \text{(iii)put t=tan x } \\ \text{9. To integrate }\quad \huge \frac{ax+b}{\sqrt{px^2+qx+r}} or \frac{ax+b}{px^2+qx+r} \\ \large express \quad ax+b=A\frac{d}{dx}(px^2+qx+r)+B \\ \text{then separate D.} \\~ \\~ \\~ \\~ \\ \huge \color{green}{\star \text{Some Shortcuts (for MCQ’s) }\star} \\ \huge \int \frac{a\sin\:x+b\cos\:x}{ c\sin\:x+d\cos\:x}=Lx+Mln|D|+c \\ \huge L=\frac{ac+bd}{c^2+d^2} \quad M=\frac{bc-ad}{ c^2+d^2} \\~ \\~ \\ \huge \int \frac{ae^x+b }{ ce^x+d}=Lx+Mln|D|+c \\ \huge L=\frac{b}{d} \quad M=\frac{a}{c}-\frac{b}{d} \\~ \\~ \\ \huge \int \frac{ae^{nx}+b }{ ce^{nx}+d}=Lx+Mln|D|+c \\ \huge L=\frac{b}{d} \quad M=\frac{1}{n}(\frac{a}{c}-\frac{b}{d})\\~ \\~ \\ \text{For partial fraction of this form, directly use} \\ \huge \frac{1}{(x+a)(x+b)}=\frac{1}{b-a}(\frac{1}{x+a}-\frac{1}{x+b})$$

2. hartnn

$\\ \text{I thought to add these hyperbolic Integrals also:} \\ \\ \large \int \sinh \: x\,dx = \cosh \:x + c \\ \large \int \cosh \:x\,dx = \sinh \:x + c \\ \large \int \tanh \:\,dx = \ln|cosh \:x| + c \\ \large \int \coth \:x\,dx = \ln|sinh \:x| + c \\ \large \int {\frac{dx}{\sqrt{a^2 + x^2}}} = \sinh ^{-1}\left( \frac{x}{a} \right) + c \\ \large \int {\frac{dx}{\sqrt{x^2 - a^2}}} =\cosh ^{-1}\left( \frac{x}{a} \right) + c \\ \large \int {\frac{dx}{a^2 - x^2}} = a^{-1}\tanh ^{-1}\left( \frac{x}{a} \right) + c; x^2 < a^2 \\ \large \int {\frac{dx}{a^2 - x^2}} = a^{-1}\coth ^{-1}\left( \frac{x}{a} \right) + c; x^2 > a^2 \\ \large \int {\frac{dx}{x\sqrt{a^2 - x^2}}} = -a^{-1} {sech}^{-1}\left( \frac{x}{a} \right) + c \\ \large \int {\frac{dx}{x\sqrt{a^2 + x^2}}} = -a^{-1} {csch}^{-1}\left| \frac{x}{a} \right| + c \\ ~$

3. RolyPoly

Is this correct? $\int tanx dx = \int \frac{sinx}{cosx} dx = -\int \frac{1}{cosx} d(cosx)$$= -\ln |cosx| +C = \ln |cosx|^{-1} +C = \ln |secx| +C$

4. hartnn

yes.

5. RolyPoly

Oh! Thanks! Actually, there are 154 formulas of indefinite integrals in my book... But it doesn't include something like 1/(asinx+bcosx+c) Thanks for sharing!!

6. hartnn

154! thats too many!

7. RolyPoly

I think so.. Formulas are basic, techniques are more important when doing integration.

8. hartnn

yes,most of these integrals can be derived using only some integrals, like you showed for tan x, so there is no need to remember all of them, just know the approach.

9. UnkleRhaukus

so how do i integrate this then \begin{align*} \\&\int\limits_0^1\sqrt{\frac{1-x^2}{1+x^2}}\cdot\text dx\\ \end{align*}

10. hartnn

@UnkleRhaukus put $$x^2=\cos^2 2\theta$$

11. UnkleRhaukus

ill try that right away

12. henpen

"Just a sec"- I see what you did there @hartnn

13. hartnn

$\begin{array}{|c|c|}\hline \text{Expression in Integral} &Substitution \\ \hline \sqrt{a^2-x^2}&x=a\sin \theta \quad or \quad x=a\cos\theta \\ \hline \sqrt{x^2-a^2}&x=a\sec \theta \quad or \quad x=a\csc\theta \\ \hline x^2+a^2 &x=a\tan \theta \quad or \quad x=a\cot\theta \\ \hline \sqrt{\frac{a-x}{a+x}}& x=a\cos2\theta \\ \hline \sqrt{\frac{a-x}{x}}or\sqrt{\frac{x}{a-x}} & x=a\sin^2\theta \\ \hline \sqrt{\frac{a+x}{x}}or\sqrt{\frac{x}{a+x}} & x=a\tan^2\theta \\ \hline \sqrt{2ax+x^2} & x=2a\tan^2\theta \\ \hline \sqrt{2ax-x^2} & x=2a\sin^2\theta \\ \hline \sqrt{\frac{a^2-x^2}{a^2+x^2}} & x^2=a^2\cos2\theta \\ \hline \end{array}$ $\begin{array}{|c|c|}\hline \text{Expression in Integral} &Substitution \\ \hline \ln|f(x)| & u=ln|f(x)| \\ \hline \ln|f(x)|\pm \ln|g(x)| & u=ln|f(x)| )|\pm \ln|g(x)| \\ \hline f(x)^nf’(x) & u=f(x) \\ \hline e^{f(x)} \quad or \quad a^{f(x)} & u=f(x) \\ \hline \sqrt{ax+b} \\ \frac{cx+d}{\sqrt{ax+b} }\$$cx+d) \sqrt{ax+b} & u= \sqrt{ax+b} \\ \hline \frac{\sin \:x+\cos \:x}{a+b\sin\:2x} & u=\int Numerator \\ \hline P(x)(ax+b)^n \\ \text{P(x)is any polynomial in x} & u=ax+b \\ \hline \frac{1}{x^{1/m}+x^(1/n)} & x=t^k,k=LCM(m,n) \\ \hline \end{array}$ 14. hartnn sorry @UnkleRhaukus \(x^2=\cos2\theta$$ so the sqrt term reduces to $$\tan \theta$$

15. hartnn

$$\int\limits_\pi^0\sqrt{\frac{1-\cos(2\theta)}{1+\cos(2\theta)}}\cdot\frac{-\sin(2\theta)}{\sqrt{\cos2\theta}}\text d\theta$$

16. hartnn

$$\int_0^\pi2\frac{\sin^2\theta}{\sqrt{\cos2\theta}}d\theta$$

17. tanjung

i like these

18. Bhagyashree

u r amazing to do tutorials like this @hartnn. it is great i mean.

19. hartnn

hey! thanks :) i went through my old integration notes and found those tips , i thought to share it here....

20. lgbasallote

21. Miyuru

Well I would like to say that nothing is clear to me.. I have never done these. :D But anyway nice tutorial @hartnn...

22. mukushla

Useful.

23. hartnn

typing mistake in 19th $$\\ \huge 19. \int \sqrt{ \frac{1}{ {x^2-a^2}}}dx=\ln|x+\sqrt{x^2-a^2}|+c \\$$

24. mayankdevnani

long work but really useful and nice work.... @hartnn

25. Zarkon

Nice work hartnn... LOL Zarkon, that looks efficient...

27. hartnn

mainly this tutorial was for tips and shortcuts, i just thought to incluse formulas and tables.....

28. hartnn

*include

29. waterineyes

Nice .. $$\huge \color{green}{^\cdot \smile^{\cdot}}$$

30. lambchamps

my goodness, you have all of this formulas in your mind?

31. gohangoku58

in his book, actually....but when he learned integration, he had all formulas in his head, because of lots of practice....

32. DLS

you can add the complex number method for integration of large powers of cosine and sine maybe :o

33. hartnn

nice suggestion, but this one was already getting to long..

34. DLS

can see D: but its awesome,is there a one for definite too?

35. hartnn

nopes, sorry.

36. AravindG

Good work :) Is there any other useful tutorial on calculus I can refer to here?

37. ikram002p

wow !

38. Sepeario

i have a question what does c mean in the results?

39. UnkleRhaukus

c is the arbitrary constant of integration ,