Question

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

Creator of Tutorial: hartnn

Answer 1

\(\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

}\)

Answer 2

\(\\~ \\~ \\~ \\~
\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})\)

Answer 3

\(\\ \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 \\ ~\)

Answer 4

Woah! Thanks for this @Tutorials!

Answer 5

No problem! The original creator of this tutorial is hartnn from OpenStudy. If you have any tutorials, please post them on QuestionCove if you want to, or we can gladly transfer it for you!