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hartnn

  • 2 years ago

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

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  1. hartnn
    • 2 years ago
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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}) \)

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

  3. RolyPoly
    • 2 years ago
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    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
    • 2 years ago
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    yes.

  5. RolyPoly
    • 2 years ago
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    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
    • 2 years ago
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    154! thats too many!

  7. RolyPoly
    • 2 years ago
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    I think so.. Formulas are basic, techniques are more important when doing integration.

  8. hartnn
    • 2 years ago
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    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
    • 2 years ago
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    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
    • 2 years ago
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    @UnkleRhaukus put \(x^2=\cos^2 2\theta \)

  11. UnkleRhaukus
    • 2 years ago
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    ill try that right away

  12. henpen
    • 2 years ago
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    "Just a sec"- I see what you did there @hartnn

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

  14. hartnn
    • 2 years ago
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    sorry @UnkleRhaukus \(x^2=\cos2\theta\) so the sqrt term reduces to \(\tan \theta\)

  15. hartnn
    • 2 years ago
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    \(\int\limits_\pi^0\sqrt{\frac{1-\cos(2\theta)}{1+\cos(2\theta)}}\cdot\frac{-\sin(2\theta)}{\sqrt{\cos2\theta}}\text d\theta\)

  16. hartnn
    • 2 years ago
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    \(\int_0^\pi2\frac{\sin^2\theta}{\sqrt{\cos2\theta}}d\theta\)

  17. tanjung
    • 2 years ago
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    i like these

  18. Bhagyashree
    • 2 years ago
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    u r amazing to do tutorials like this @hartnn. it is great i mean.

  19. hartnn
    • 2 years ago
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    hey! thanks :) i went through my old integration notes and found those tips , i thought to share it here....

  20. lgbasallote
    • 2 years ago
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    they're back....the dreaded tutorials.....

  21. Miyuru
    • 2 years ago
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    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
    • 2 years ago
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    Useful.

  23. hartnn
    • 2 years ago
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    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
    • 2 years ago
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    long work but really useful and nice work.... @hartnn

  25. Zarkon
    • 2 years ago
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    how about just posting a link? http://integral-table.com/

  26. Shadowys
    • 2 years ago
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    Nice work hartnn... LOL Zarkon, that looks efficient...

  27. hartnn
    • 2 years ago
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    mainly this tutorial was for tips and shortcuts, i just thought to incluse formulas and tables.....

  28. hartnn
    • 2 years ago
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    *include

  29. waterineyes
    • 2 years ago
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    Nice .. \(\huge \color{green}{^\cdot \smile^{\cdot}}\)

  30. lambchamps
    • 2 years ago
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    my goodness, you have all of this formulas in your mind?

  31. gohangoku58
    • 2 years ago
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    in his book, actually....but when he learned integration, he had all formulas in his head, because of lots of practice....

  32. DLS
    • one year ago
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    you can add the complex number method for integration of large powers of cosine and sine maybe :o

  33. hartnn
    • one year ago
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    nice suggestion, but this one was already getting to long..

  34. DLS
    • one year ago
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    can see D: but its awesome,is there a one for definite too?

  35. hartnn
    • one year ago
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    nopes, sorry.

  36. AravindG
    • 8 months ago
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    Good work :) Is there any other useful tutorial on calculus I can refer to here?

  37. ikram002p
    • 4 months ago
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    wow !

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