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hartnn Group Title

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

  • one year ago
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  1. hartnn Group Title
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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}) \)

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

    • one year ago
  3. RolyPoly Group Title
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    Is this correct? \[\int tanx dx = \int \frac{sinx}{cosx} dx = -\int \frac{1}{cosx} d(cosx)\]\[ = -\ln |cosx| +C = \ln |cosx|^{-1} +C = \ln |secx| +C\]

    • one year ago
  4. hartnn Group Title
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    yes.

    • one year ago
  5. RolyPoly Group Title
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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!!

    • one year ago
  6. hartnn Group Title
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    154! thats too many!

    • one year ago
  7. RolyPoly Group Title
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    I think so.. Formulas are basic, techniques are more important when doing integration.

    • one year ago
  8. hartnn Group Title
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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.

    • one year ago
  9. UnkleRhaukus Group Title
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    so how do i integrate this then \[\begin{align*} \\&\int\limits_0^1\sqrt{\frac{1-x^2}{1+x^2}}\cdot\text dx\\ \end{align*}\]

    • one year ago
  10. hartnn Group Title
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    @UnkleRhaukus put \(x^2=\cos^2 2\theta \)

    • one year ago
  11. UnkleRhaukus Group Title
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    ill try that right away

    • one year ago
  12. henpen Group Title
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    "Just a sec"- I see what you did there @hartnn

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

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

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

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

    • one year ago
  17. tanjung Group Title
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    i like these

    • one year ago
  18. Bhagyashree Group Title
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    u r amazing to do tutorials like this @hartnn. it is great i mean.

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

    • one year ago
  20. lgbasallote Group Title
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    they're back....the dreaded tutorials.....

    • one year ago
  21. Miyuru Group Title
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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...

    • one year ago
  22. mukushla Group Title
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    Useful.

    • one year ago
  23. hartnn Group Title
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    typing mistake in 19th \(\\ \huge 19. \int \sqrt{ \frac{1}{ {x^2-a^2}}}dx=\ln|x+\sqrt{x^2-a^2}|+c \\ \)

    • one year ago
  24. mayankdevnani Group Title
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    long work but really useful and nice work.... @hartnn

    • one year ago
  25. Zarkon Group Title
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    how about just posting a link? http://integral-table.com/

    • one year ago
  26. Shadowys Group Title
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    Nice work hartnn... LOL Zarkon, that looks efficient...

    • one year ago
  27. hartnn Group Title
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    mainly this tutorial was for tips and shortcuts, i just thought to incluse formulas and tables.....

    • one year ago
  28. hartnn Group Title
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    *include

    • one year ago
  29. waterineyes Group Title
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    Nice .. \(\huge \color{green}{^\cdot \smile^{\cdot}}\)

    • one year ago
  30. lambchamps Group Title
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    my goodness, you have all of this formulas in your mind?

    • one year ago
  31. gohangoku58 Group Title
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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....

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

    • 10 months ago
  33. hartnn Group Title
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    nice suggestion, but this one was already getting to long..

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

    • 10 months ago
  35. hartnn Group Title
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    nopes, sorry.

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

    • 4 months ago
  37. ikram002p Group Title
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    wow !

    • 21 days ago
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