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

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\( \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}) \)
\[\\ \text{I thought to add these hyperbolic Integrals also:} \\ \\ \large \int \sinh \: x\,dx = \cosh \:x + c \\ \large \int \cosh \:x\,dx = \sinh \:x + c \\ \large \int \tanh \:\,dx = \ln|cosh \:x| + c \\ \large \int \coth \:x\,dx = \ln|sinh \:x| + c \\ \large \int {\frac{dx}{\sqrt{a^2 + x^2}}} = \sinh ^{-1}\left( \frac{x}{a} \right) + c \\ \large \int {\frac{dx}{\sqrt{x^2 - a^2}}} =\cosh ^{-1}\left( \frac{x}{a} \right) + c \\ \large \int {\frac{dx}{a^2 - x^2}} = a^{-1}\tanh ^{-1}\left( \frac{x}{a} \right) + c; x^2 < a^2 \\ \large \int {\frac{dx}{a^2 - x^2}} = a^{-1}\coth ^{-1}\left( \frac{x}{a} \right) + c; x^2 > a^2 \\ \large \int {\frac{dx}{x\sqrt{a^2 - x^2}}} = -a^{-1} {sech}^{-1}\left( \frac{x}{a} \right) + c \\ \large \int {\frac{dx}{x\sqrt{a^2 + x^2}}} = -a^{-1} {csch}^{-1}\left| \frac{x}{a} \right| + c \\ ~ \]
Is this correct? \[\int tanx dx = \int \frac{sinx}{cosx} dx = -\int \frac{1}{cosx} d(cosx)\]\[ = -\ln |cosx| +C = \ln |cosx|^{-1} +C = \ln |secx| +C\]

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yes.
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!!
154! thats too many!
I think so.. Formulas are basic, techniques are more important when doing integration.
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.
so how do i integrate this then \[\begin{align*} \\&\int\limits_0^1\sqrt{\frac{1-x^2}{1+x^2}}\cdot\text dx\\ \end{align*}\]
@UnkleRhaukus put \(x^2=\cos^2 2\theta \)
ill try that right away
"Just a sec"- I see what you did there @hartnn
\[\begin{array}{|c|c|}\hline \text{Expression in Integral} &Substitution \\ \hline \sqrt{a^2-x^2}&x=a\sin \theta \quad or \quad x=a\cos\theta \\ \hline \sqrt{x^2-a^2}&x=a\sec \theta \quad or \quad x=a\csc\theta \\ \hline x^2+a^2 &x=a\tan \theta \quad or \quad x=a\cot\theta \\ \hline \sqrt{\frac{a-x}{a+x}}& x=a\cos2\theta \\ \hline \sqrt{\frac{a-x}{x}}or\sqrt{\frac{x}{a-x}} & x=a\sin^2\theta \\ \hline \sqrt{\frac{a+x}{x}}or\sqrt{\frac{x}{a+x}} & x=a\tan^2\theta \\ \hline \sqrt{2ax+x^2} & x=2a\tan^2\theta \\ \hline \sqrt{2ax-x^2} & x=2a\sin^2\theta \\ \hline \sqrt{\frac{a^2-x^2}{a^2+x^2}} & x^2=a^2\cos2\theta \\ \hline \end{array} \] \[\begin{array}{|c|c|}\hline \text{Expression in Integral} &Substitution \\ \hline \ln|f(x)| & u=ln|f(x)| \\ \hline \ln|f(x)|\pm \ln|g(x)| & u=ln|f(x)| )|\pm \ln|g(x)| \\ \hline f(x)^nf’(x) & u=f(x) \\ \hline e^{f(x)} \quad or \quad a^{f(x)} & u=f(x) \\ \hline \sqrt{ax+b} \\ \frac{cx+d}{\sqrt{ax+b} }\\(cx+d) \sqrt{ax+b} & u= \sqrt{ax+b} \\ \hline \frac{\sin \:x+\cos \:x}{a+b\sin\:2x} & u=\int Numerator \\ \hline P(x)(ax+b)^n \\ \text{P(x)is any polynomial in x} & u=ax+b \\ \hline \frac{1}{x^{1/m}+x^(1/n)} & x=t^k,k=LCM(m,n) \\ \hline \end{array} \]
sorry @UnkleRhaukus \(x^2=\cos2\theta\) so the sqrt term reduces to \(\tan \theta\)
\(\int\limits_\pi^0\sqrt{\frac{1-\cos(2\theta)}{1+\cos(2\theta)}}\cdot\frac{-\sin(2\theta)}{\sqrt{\cos2\theta}}\text d\theta\)
\(\int_0^\pi2\frac{\sin^2\theta}{\sqrt{\cos2\theta}}d\theta\)
i like these
u r amazing to do tutorials like this @hartnn. it is great i mean.
hey! thanks :) i went through my old integration notes and found those tips , i thought to share it here....
they're back....the dreaded tutorials.....
Well I would like to say that nothing is clear to me.. I have never done these. :D But anyway nice tutorial @hartnn...
Useful.
typing mistake in 19th \(\\ \huge 19. \int \sqrt{ \frac{1}{ {x^2-a^2}}}dx=\ln|x+\sqrt{x^2-a^2}|+c \\ \)
long work but really useful and nice work.... @hartnn
how about just posting a link? http://integral-table.com/
Nice work hartnn... LOL Zarkon, that looks efficient...
mainly this tutorial was for tips and shortcuts, i just thought to incluse formulas and tables.....
*include
Nice .. \(\huge \color{green}{^\cdot \smile^{\cdot}}\)
my goodness, you have all of this formulas in your mind?
in his book, actually....but when he learned integration, he had all formulas in his head, because of lots of practice....
  • DLS
you can add the complex number method for integration of large powers of cosine and sine maybe :o
nice suggestion, but this one was already getting to long..
  • DLS
can see D: but its awesome,is there a one for definite too?
nopes, sorry.
Good work :) Is there any other useful tutorial on calculus I can refer to here?
wow !
i have a question what does c mean in the results?
c is the arbitrary constant of integration ,

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