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Index of Indefinite Integration

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Pattern – 1 : Wrong interpretation of basic formula \int\frac{1}{x^2+16}dx=\ \int\frac{1}{\left(x\right)^2+\left(4\right)^2}\ dx

Pattern – 2 :f ∫f(x) dx = F(x) +c, then ∫ f (a + b) dx = \frac{1}{a} F ( ax + b )dx + c  :  a ≠ 0

pattern – 3 : Adjustment  \int\frac{2x^3+18x-1}{x^2+9}\ dx

Pattern – 4 : ∫ \left[f\left(x\right)\right]^n f' (x) dx = \frac{\left[f\left(x\right)\right]^{n+1\ }}{n+1}+c

Pattern – 5 :\int\frac{f\prime\left(x\right)}{f\left(x\right)}\ dx=\log{\left|f\left(x\right)\right|}+c

Pattern – 6 : Method of substitution 1)\ \ \int\frac{x^2}{\sqrt{x^6-a^6}}\ dx

Pattern – 7 : \frac{1}{ax^2+bx+c}, \ \frac{1}{\sqrt{ax^2+bx+c}}, \ \frac{Ax+B}{ax^2+bx+c}, \ \frac{Ax+B}{\sqrt{ax^2+bx+c}}

Pattern – 8 : Trigonometric substitution

Pattern – 9 : \frac{1}{a\cos{x+b\sin{x}+c}}\ or\ \frac{1}{a\cos{x}+b\sin{x}}\ or\ \frac{1}{a\cos{x\ }+b}\ or\ \frac{1}{a\sin{x}+b}

Pattern – 10 : ∫ \sin^n{x}\times \cos^m{x} dx

Pattern – 11 \frac{1}{a\cos^2{x}+b\sin^2{x}+c}\ or\ \frac{1}{a\cos^2{x\ }+b}\ or\ \frac{1}{a\sin^2{x}+b}\ or\ \frac{1}{a\cos^2{x}+b\sin^2{x}}

Pattern – 12 \frac{A\sin{x}\times \cos{x}}{a\cos^2{x}+b\sin^2{x}+c}\ OR\ \ \frac{A\tan{x}}{a\tan^2{x}+b\sec^2{x}+c}

Pattern – 13 Integration by parts ∫ uv dx = u ∫ vdx – ∫\left(\frac{du}{dx}\ \int v\ dx\right) dx

Pattern – 14 ( Some special formulas )

Pattern – 15  ∫ e^{ax} sin (bx + k) dx

Pattern – 16 ∫ e^{ax+b} [ a f(x) + f' (x) ] dx = e^{ax+b} f(x) + c

Pattern – 17 I=\ \int\frac{p\left(x\right)}{q\left(x\right)}\ \ dx , where\ d ( p ( x ) ) < d ( q ( x )) Where, q ( x ) = \left(a_1\ x-b_1\right)\ \left(a_2\ x-b_2\right)\times\ldots\times\left(a_n\ x-bn\ \right)

Pattern – 18 I=\ \int\frac{p\left(x\right)}{q\left(x\right)}\ dx_1\ where, d ( p ( x ) ) < d ( q ( x ) )

Pattern – 19 I =\ \int\frac{p\left(x\right)}{q\left(x\right)}\ dx_1    where, d ( p ( x ) ) < d ( q ( x ) )

Pattern – 20 Short trick \frac{1}{\left(a_1x-b_1\right)\left(a_2x-b_2\right)}\ OR\ \frac{x}{\left(a_1x-b_1\right)\left(a_2x-b_2\right)}

Pattern – 21 \int\frac{1}{\left(ax+b\right)^m\ \left(cx+d\right)^n}dx; where, m+n=2

Pattern – 22  I=\ \int\frac{1}{\sin^\frac{3}{2}{x}\times \cos^\frac{1}{2}{x}}\ dx

Pattern – 23 \int\frac{1}{\sin^2{x}\times \cos^2{x}}\ dx,\ \int\frac{1}{\sin{x}\times \cos^2{x}\ }dx,

Pattern – 24 \int\frac{p\cos{x}+q\sin{x}+r}{a\cos{x}+b\sin{x}+n}\ dx

Pattern – 25 \int\frac{ae^x+be^{-x}+n}{pe^x+qe^{-x}+r}\ dx where, a, b , n, p, q, r ∈ R then use

Pattern – 26 \int\frac{x^2+a}{x^4+bx^2+c\ }\ dx then divide numerator & denominator by  & assume x\pm\frac{k}{x}=t

Pattern – 27 If I_n = ∫\tan^n{x} dx, then I_n=\ \frac{\tan^{n-1\ }{x}}{n-1}-I_n-2\ :n\geq2

Pattern – 28  ∫ \cot^n{x} dx, then I_n=\ \frac{-\cot^{n-1\ }{x}}{x-1}-I_{n-2}\ :n\geq2

Pattern – 29 ∫ \sec^n{x} dx , thenI_n=\ \frac{\tan{x\times \sec^{n-2}{x}}}{n-1}+\frac{n-2}{n-1}\ I_{n-2}\ :n\geq2

Pattern – 30 :  I_n=\ \int{\rm cosec}^n\ x\ dx

Pattern – 31 : I_n=\ \int\sin^n{x}\ dx

Pattern – 32 : I_n=\cos^n{x} dx

Pattern – 33 : I_n=\ \int x^n sin\ mx\ dx, then I_n =\ \frac{x^n\cos{mx}}{m}+\ \frac{nx^{n-1}\sin{mx}}{m^2}-\ \frac{n\left(n-1\right)}{m^2}I_{n-2} : n ≥ 2

Pattern – 34 : I_n=\ \int x^n sin mx dx, then I_n=\ \frac{x^n\sin{mx}}{m}+\ \frac{nx^{n-1}\cos{mx}}{m^2} – \ \frac{n\left(n-1\right)}{m^2}I_{n-2} : n ≥ 2

Pattern – 35 : I_n=\ \int x^n\ e^{ax}\ dx

Pattern – 36 : I_n=\ \int\left(\log{x}\right)^n\ dx

Pattern – 37 : \int\frac{1}{\sqrt{\left(x-a\right)\left(b-x\right)}}dx,\ \int\sqrt{\frac{x-a}{b-x}}\ dx,  then  put x = a cos^2{\theta\ } + b \sin^2{\theta\ }

Pattern – 38 : \int\frac{1}{\left(px+q\right)\sqrt{ax+b}}dx, then put ax + b + t^2

Pattern – 39 : \int\frac{ax+b}{\left(px^2+qx+r\right)\ \sqrt{mx+n\ }}dx 

Pattern – 40 : \int\frac{1}{\left(px+q\ \right)\sqrt{ax^2+bx+c}}\ dx, then put px+q=\frac{1}{t}

Pattern : 41 : \int\frac{1}{\left(px^2+q\right)\sqrt{ax^2+b}}dx,  then put x= \frac{1}{t} , then put a+bt^2=z^2

Pattern – 42 : \int\frac{ax^2+bx+c}{\left(qx+4\right)\sqrt{px^2+1x+r}}\ dx

Pattern – 43(i) : Integrals of differential binomials and chebyslev’s  criterion

Pattern – 43 (ii) : Integrals of differential binomials and chebyslev’s  criterion ∫dx  : p ≠ 0

Pattern –  43(iii) : Integrals of differential binomials and chebyslev’s  criterion \int x^m\left(a+bx^n\right)^p\ dx  : p ≠ 0

Pattern – 43 (iv) : Integrals of differential binomials and chebyslev’s  criterion \int x^m\left(a+bx^n\right)^p\ dx  : p ≠ 0

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