Integration Mathematical Formula

Integration Mathematical Formula :

1. Common Integrals
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—-Indefinite Integral——–

\int f\left ( g\left ( x \right ) \right )g'\left ( x \right )dx = \int f\left ( u \right )du

 

\int f\left ( x \right )g'\left ( x \right )dx = f\left ( x \right )g\left ( x \right ) - \int g\left ( x \right )f'\left ( x \right )dx

 

 

——-Integrals of Rational and Irrational Functions——-

\int x^{n}dx = \frac{x^{n+1}}{n+1}+C

 

\int \frac{1}{x}dx = ln\left | x \right |+C

 

\int c dx = cx+C

 

\int x dx = \frac{x^{2}}{2}+C

 

\int x^{2} dx = \frac{x^{3}}{3}+C

 

\int \frac{1}{x^{2}} dx = -\frac{1}{x}+C

 

\int \sqrt{x} dx = \frac{2x\sqrt{x}}{3}+C

 

\int \frac{1}{1+x^{2}} dx = arctan x+C

 

\int \frac{1}{\sqrt{1-x^{2}}} dx = arcsin x+C

 

 

——-Integrals of Trigonometric Functions——

\int sin \hspace{.1cm} x \hspace{.1cm} dx = -cos \hspace{.1cm} x+C

 

\int cos \hspace{.1cm} x \hspace{.1cm} dx = sin \hspace{.1cm} x+C

 

\int tan \hspace{.1cm} x \hspace{.1cm} dx = ln \left | sec \hspace{.1cm} x \right |+C

 

\int sec \hspace{.1cm} x \hspace{.1cm} dx = ln \left | tan \hspace{.1cm} x + sec \hspace{.1cm} x \right |+C

 

\int sin^{2} \hspace{.1cm} x \hspace{.1cm} dx = \frac{1}{2}\left ( x-sin \hspace{.1cm} x \hspace{.1cm} cos \hspace{.1cm} x \right )+C

 

\int cos^{2} \hspace{.1cm} x \hspace{.1cm} dx = \frac{1}{2}\left ( x+sin \hspace{.1cm} x \hspace{.1cm} cos \hspace{.1cm} x \right )+C

 

\int tan^{2} \hspace{.1cm} x \hspace{.1cm} dx = tan \hspace{.1cm} x-x+C

 

\int sec^{2} \hspace{.1cm} x \hspace{.1cm} dx = tan \hspace{.1cm} x+C

 

 

———Integrals of Exponential and Logarithmic Functions———

\int ln \hspace{.1cm} x \hspace{.1cm} dx = x \hspace{.1cm} ln \hspace{.1cm} x-x+C

 

\int x^{n}ln \hspace{.1cm} x \hspace{.1cm} dx = \frac{x^{n+1}}{n+1} ln \hspace{.1cm} x-\frac{x^{n+1}}{\left ( n+1 \right )^{2}}+C

 

\int e^{x} dx = e^{x}+C

 

\int b^{x} dx = \frac{b^{x}}{ln \hspace{.1cm} b}+C

 

\int b^{x} dx = \frac{b^{x}}{ln \hspace{.1cm} b}+C

 

\int sinh \hspace{.1cm} x \hspace{.1cm} dx = cosh \hspace{.1cm} x+C

 

\int cosh \hspace{.1cm} x \hspace{.1cm} dx = sinh \hspace{.1cm} x+C

 

 

 

2. Integrals of Rational Functions
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——-Integrals involving ax + b——-

\int \left ( ax+b \right )^{n}dx = \frac{\left ( ax+b \right )^{n+1}}{a\left ( n+1 \right )} \hspace{.1cm} \hspace{.1cm} \left (for \hspace{.1cm} n\neq -1 \right )

 

\int \frac{1}{ax+b}dx = \frac{1}{a}ln\left | ax+b \right |

 

\int x\left ( ax+b \right )^{n}dx = \frac{a\left ( n+1 \right )x-b}{a^{2}\left ( n+1 \right )\left ( n+2 \right )}\left ( ax+b \right )^{n+1} \hspace{.1cm} \hspace{.1cm} \left (for \hspace{.1cm} n\neq -1, n\neq -2 \right )

 

\int \frac{x}{ax+b}dx = \frac{x}{a}-\frac{b}{a^{2}}ln\left | ax+b \right |

 

\int \frac{x}{\left ( ax+b \right )^{2}}dx = \frac{b}{a^{2}\left ( ax+b \right )}+\frac{1}{a^{2}}ln\left | ax+b \right |

 

\int \frac{x}{\left ( ax+b \right )^{n}}dx = \frac{a\left ( 1-n \right )x-b}{a^{2}\left ( n-1 \right )\left ( n-2 \right )\left ( ax+b \right )^{n-1}} \hspace{.1cm} \hspace{.1cm} \left (for \hspace{.1cm} n\neq -1, n\neq -2 \right )

 

\int \frac{x^{2}}{ax+b}dx = \frac{1}{a^{3}}\left ( \frac{\left ( ax+b \right )^{2}}{2}-2b\left ( ax+b \right )+b^{2}ln\left | ax+b \right | \right )

 

\int \frac{x^{2}}{\left ( ax+b \right )^{2}}dx = \frac{1}{a^{3}}\left ( ax+b-2b \hspace{.1cm} ln\left | ax+b \right |-\frac{b^{2}}{ax+b} \right )

 

\int \frac{x^{2}}{\left ( ax+b \right )^{3}}dx = \frac{1}{a^{3}}\left ( ln\left | ax+b \right |+\frac{2b}{ax+b}-\frac{b^{2}}{2\left ( ax+b \right )^{2}} \right )

 

\int \frac{x^{2}}{\left ( ax+b \right )^{n}}dx = \frac{1}{a^{3}}\left ( -\frac{\left ( ax+b \right )^{3-n}}{n-3}+\frac{2b\left ( a+b \right )^{2-n}}{n-2}-\frac{b^{2}\left ( ax+b \right )^{1-n}}{n-1} \right ) \hspace{.1cm} \hspace{.1cm} \left (for \hspace{.1cm} n\neq 1,2,3 \right )

 

\int \frac{1}{x\left ( ax+b \right )}dx = -\frac{1}{b}ln\left | \frac{ax+b}{x} \right |

 

\int \frac{1}{x^{2}\left ( ax+b \right )}dx = -\frac{1}{bx}+\frac{a}{b^{2}}ln\left | \frac{ax+b}{x} \right |

 

\int \frac{1}{x^{2}\left ( ax+b \right )^{2}}dx = -a\left ( \frac{1}{b^{2}\left ( a+xb \right )}+\frac{1}{ab^{2}x}-\frac{2}{b^{3}}ln\left | \frac{ax+b}{x} \right | \right )

 

 

———Integrals involving ax2 + bx + c——

\int \frac{1}{x^{2}+a^{2}}dx = \frac{1}{a}arctg\frac{x}{a}

 

\int \frac{1}{x^{2}-a^{2}}dx = \left\{\begin{matrix} \frac{1}{2a}ln\frac{a-x}{a+x} & \hspace{.1cm} \hspace{.1cm} for\left | x \right |<\left | a \right |\\ \frac{1}{2a}ln\frac{x-a}{x+a} & \hspace{.1cm} \hspace{.1cm} for\left | x \right |>\left | a \right | \end{matrix}\right.

 

\int \frac{1}{ax^{2}+bx+c}dx =\left\{\begin{matrix} \frac{2}{\sqrt{4ac-b^{2}}}arctan\frac{2ax+b}{\sqrt{4ac-b^{2}}} & \hspace{.1cm} \hspace{.1cm} for \hspace{.1cm} 4ac-b^{2}>0 \\ \frac{2}{\sqrt{b^{2}-4ac}}ln\left | \frac{2ax+b-\sqrt{b^{2}-4ac}}{2ax+b+\sqrt{b^{2}-4ac}} \right | & \hspace{.1cm} \hspace{.1cm} for \hspace{.1cm} 4ac-b^{2}<0 \\ -\frac{2}{2ax+b} & \hspace{.1cm} \hspace{.1cm} for \hspace{.1cm} 4ac-b^{2}=0 \end{matrix}\right.

 

\int \frac{x}{ax^{2}+bx+c}dx = \frac{1}{2a}ln\left | ax^{2}+bx+c \right |-\frac{b}{2a}\int \frac{dx}{ax^{2}+bx+c}

 

\int \frac{mx+n}{ax^{2}+bx+c}dx = \left\{\begin{matrix} \frac{m}{2a}ln\left | ax^{2}+bx+c \right |+\frac{2an-bm}{a\sqrt{4ac-b^{2}}}arctan\frac{2ax+b}{\sqrt{4ac-b^{2}}} & \hspace{.1cm} \hspace{.1cm} for \hspace{.1cm} 4ac-b^{2}>0\\ \frac{m}{2a}ln\left | ax^{2}+bx+c \right |+\frac{2an-bm}{a\sqrt{b^{2}-4ac}}arctanh\frac{2ax+b}{\sqrt{b^{2}-4ac}} & \hspace{.1cm} \hspace{.1cm} for \hspace{.1cm} 4ac-b^{2}<0\\ \frac{m}{2a}ln\left | ax^{2}+bx+c \right |-\frac{2an-bm}{a\left ( 2ax+b \right )} & \hspace{.1cm} \hspace{.1cm} for \hspace{.1cm} 4ac-b^{2}=0 \end{matrix}\right.

 

\int \frac{1}{\left ( ax^{2}+bx+c \right )^{n}}dx = \frac{2ax+b}{\left ( n-1 \right )\left ( 4ac-b^{2} \right )\left ( ax^{2}+bx+c \right )^{n-1}}+\frac{\left ( 2n-3 \right )2a}{\left ( n-1 \right )\left ( 4ac-b^{2} \right )}\int \frac{1}{\left ( ax^{2}+bx+c \right )^{n-1}}dx

 

\int \frac{1}{x\left ( ax^{2}+bx+c \right )}dx = \frac{1}{2c}ln\left | \frac{x^{2}}{ax^{2}+bx+c} \right |-\frac{b}{2c}\int \frac{1}{ax^{2}+bx+c} dx

 

 

 

3. Integrals of Exponential Functions
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\int xe^{cx}dx = \frac{e^{cx}}{c^{2}}\left ( cx-1 \right )

 

\int x^{2}e^{cx}dx = e^{cx}\left ( \frac{x^{2}}{c}-\frac{2x}{c^{2}}+\frac{2}{c^{3}} \right )

 

\int x^{n}e^{cx}dx = \frac{1}{c}x^{n}e^{cx}-\frac{n}{c}\int x^{n-1}e^{cx}dx

 

\int \frac{e^{cx}}{x}dx = ln\left | x \right |+\sum\limits_{i=1}^{\infty }\frac{\left ( cx \right )^{i}}{i\cdot i!}

 

\int e^{cx}ln \hspace{.1cm} xdx = \frac{1}{c}e^{cx}ln\left | x \right |+E_{i}\left ( cx \right )

 

\int e^{cx}sin \hspace{.1cm} bxdx = \frac{e^{cx}}{c^{2}+b^{2}}\left ( c \hspace{.1cm} sin \hspace{.1cm} bx-b \hspace{.1cm} cos \hspace{.1cm} bx \right )

 

\int e^{cx}cos \hspace{.1cm} bxdx = \frac{e^{cx}}{c^{2}+b^{2}}\left ( c \hspace{.1cm} cos \hspace{.1cm} bx+b \hspace{.1cm} sin \hspace{.1cm} bx \right )

 

\int e^{cx}sin^{n} \hspace{.1cm} xdx = \frac{e^{cx} sin^{n-1}x}{c^{2}+n^{2}}\left ( c \hspace{.1cm} sin \hspace{.1cm} x-n \hspace{.1cm} cos \hspace{.1cm} bx \right )+\frac{n\left ( n-1 \right )}{c^{2}+n^{2}}\int e^{cx}sin^{n-2} \hspace{.1cm} dx

 

 

 

4. Integrals of Logarithmic Functions
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\int ln \hspace{.1cm} cxdx = x \hspace{.1cm} ln \hspace{.1cm} cx-x

 

\int ln\left ( ax+b \right )dx = x \hspace{.1cm} ln\left ( ax+b \right ) - x + \frac{b}{a}ln\left ( ax+b \right )

 

\int \left ( ln \hspace{.1cm} x \right )^{2}dx = x\left ( ln \hspace{.1cm} x \right )^{2} - 2x \hspace{.1cm} ln \hspace{.1cm} x+2x

 

\int \left ( ln \hspace{.1cm} cx \right )^{n}dx = x\left ( ln \hspace{.1cm} cx \right )^{n} - n\int \left ( ln \hspace{.1cm} cx \right )^{n-1}dx

 

\int \frac{dx}{ln \hspace{.1cm} x} = ln\left | ln \hspace{.1cm} x \right | + ln \hspace{.1cm} x + \sum\limits_{n=2}^{\infty }\frac{\left ( ln \hspace{.1cm} x \right )^{i}}{i\cdot i!}

 

\int \frac{dx}{\left (ln \hspace{.1cm} x \right )^{n}} = - \frac{x}{\left ( n-1 \right )\left ( ln \hspace{.1cm} x \right )^{n-1}} + \frac{1}{n-1}\int \frac{dx}{\left ( ln \hspace{.1cm} x \right )^{n-1}} \hspace{.1cm} \hspace{.1cm} \left ( for \hspace{.1cm} n\neq 1 \right )

 

\int x^{m}ln \hspace{.1cm} xdx = x^{m+1}\left ( \frac{ln \hspace{.1cm} x}{m+1} - \frac{1}{\left ( m+1 \right )^{2}} \right ) \hspace{.1cm} \hspace{.1cm} \left ( for \hspace{.1cm} m\neq 1 \right )

 

\int x^{m}\left (ln \hspace{.1cm} x \right )^{n}dx = \frac{x^{m+1}\left (ln \hspace{.1cm} x \right )^{n}}{m+1} \hspace{.1cm}-\frac{n}{m+1}\int x^{m}\left ( ln \hspace{.1cm} x \right )^{n-1} \hspace{.1cm} \left ( for \hspace{.1cm} m\neq 1 \right )

 

\int \frac{\left (ln \hspace{.1cm} x \right )^{n}}{x}dx = \frac{\left (ln \hspace{.1cm} x \right )^{n+1}}{n+1} \hspace{.1cm} \left ( for \hspace{.1cm} n\neq 1 \right )

 

\int \frac{ln \hspace{.1cm} x^{n}}{x}dx = \frac{\left (ln \hspace{.1cm} x^{n} \right )^{2}}{2n} \hspace{.1cm} \left ( for \hspace{.1cm} n\neq 0 \right )

 

\int \frac{ln \hspace{.1cm} x}{x^{m}}dx = -\frac{ln \hspace{.1cm} x}{\left ( m-1 \right )x^{m-1}}-\frac{1}{\left ( m-1 \right )^{2}x^{m-1}} \hspace{.1cm} \left ( for \hspace{.1cm} m\neq 1 \right )

 

\int \frac{\left (ln \hspace{.1cm} x \right )^{n}}{x^{m}}dx = -\frac{\left (ln \hspace{.1cm} x \right )^{n}}{\left ( m-1 \right )x^{m-1}}+\frac{n}{m-1}\int \frac{\left (ln \hspace{.1cm} x \right )^{n-1}}{x^{m}}dx \hspace{.1cm} \left ( for \hspace{.1cm} m\neq 1 \right )

 

\int \frac{dx}{x \hspace{.1cm} ln \hspace{.1cm} x} = ln\left | ln \hspace{.1cm} x \right |

 

\int \frac{dx}{x^{n} \hspace{.1cm} ln \hspace{.1cm} x} = ln\left | ln \hspace{.1cm} x \right | + \sum\limits_{i=1}^{\infty }\left ( -1 \right )^{i}\frac{\left ( n-1 \right )^{i}\left ( ln \hspace{.1cm} x \right )^{i}}{i\cdot i!}

 

\int \frac{dx}{x \hspace{.1cm} \left (ln \hspace{.1cm} x \right )^{n}} = -\frac{1}{\left ( n-1 \right )\left ( ln \hspace{.1cm} x \right )^{n-1}} \hspace{.1cm} \left ( for \hspace{.1cm} n\neq 1 \right )

 

\int ln \left ( x^{2}+a^{2} \right )dx = x \hspace{.1cm} ln\left ( x^{2}+a^{2} \right )-2x+2a \hspace{.1cm} tan^{-1}\frac{x}{a}

 

\int sin\left ( ln \hspace{.1cm} x \right )dx = \frac{x}{2}\left ( sin\left ( ln \hspace{.1cm} x \right )-cos\left ( ln \hspace{.1cm} x \right ) \right )

 

\int cos\left ( ln \hspace{.1cm} x \right )dx = \frac{x}{2}\left ( sin\left ( ln \hspace{.1cm} x \right )+cos\left ( ln \hspace{.1cm} x \right ) \right )

 

 

 

5. Integrals of Trig. Functions
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\int sin \hspace{.1cm} xdx = -cos \hspace{.1cm} x

 

\int cos \hspace{.1cm} xdx = -sin \hspace{.1cm} x

 

\int sin^{2} \hspace{.1cm} xdx = \frac{x}{2}-\frac{1}{4}sin \hspace{.1cm} 2x

 

\int cos^{2} \hspace{.1cm} xdx = \frac{x}{2}+\frac{1}{4}sin \hspace{.1cm} 2x

 

\int sin^{3} \hspace{.1cm} xdx = \frac{1}{3}cos^{3}x-cos \hspace{.1cm} x

 

\int cos^{3} \hspace{.1cm} xdx = sin \hspace{.1cm} x-\frac{1}{3}sin^{3}x

 

\int \frac{dx}{sin \hspace{.1cm} x}xdx = ln\left | tan\frac{x}{2} \right |

 

\int \frac{dx}{cos \hspace{.1cm} x}xdx = ln\left | tan\left (\frac{x}{2}+\frac{\Pi }{4} \right ) \right |

 

\int \frac{dx}{sin^{2}x}xdx = -cot \hspace{.1cm} x

 

\int \frac{dx}{cos^{2}x}xdx = tan \hspace{.1cm} x

 

\int \frac{dx}{sin^{3}x} = -\frac{cos \hspace{.1cm} x}{2sin^{2}x}+\frac{1}{2}ln\left | tan\frac{x}{2} \right |

 

\int \frac{dx}{cos^{3}x} = \frac{sin \hspace{.1cm} x}{2cos^{2}x}+\frac{1}{2}ln\left | tan\left (\frac{x}{2}+\frac{\Pi }{4} \right ) \right |

 

\int sin \hspace{.1cm} x \hspace{.1cm} cos \hspace{.1cm} xdx = -\frac{1}{4}cos \hspace{.1cm} 2x

 

\int sin^{2} \hspace{.1cm} x \hspace{.1cm} cos \hspace{.1cm} xdx = \frac{1}{3}sin^{3} \hspace{.1cm} x

 

\int sin \hspace{.1cm} x \hspace{.1cm} cos^{2} \hspace{.1cm} xdx = -\frac{1}{3}cos^{3} \hspace{.1cm} x

 

\int sin^{2} \hspace{.1cm} x \hspace{.1cm} cos^{2} \hspace{.1cm} xdx = \frac{x}{8}-\frac{1}{32}sin \hspace{.1cm} 4x

 

\int tan \hspace{.1cm} xdx = -ln\left | cos \hspace{.1cm} x \right |

 

\int \frac{sin \hspace{.1cm} x}{cos^{2} \hspace{.1cm} x}dx = \frac{1}{cos \hspace{.1cm} x}

 

\int \frac{sin^{2} \hspace{.1cm} x}{cos \hspace{.1cm} x}dx = ln\left | tan\left ( \frac{x}{2}+\frac{\Pi }{4} \right ) \right |-sin \hspace{.1cm} x

 

\int tan^{2} \hspace{.1cm} xdx = tan \hspace{.1cm} x-x

 

\int cot \hspace{.1cm} xdx = ln\left | sin \hspace{.1cm} x \right |

 

\int \frac{cos \hspace{.1cm} x}{sin^{2} \hspace{.1cm} x}dx = -\frac{1}{sin \hspace{.1cm} x}

 

\int \frac{cos^{2} \hspace{.1cm} x}{sin \hspace{.1cm} x}dx = ln\left | tan\frac{x}{2} \right |+cos \hspace{.1cm} x

 

\int cot^{2} \hspace{.1cm} xdx = -cot \hspace{.1cm} x-x

 

\int \frac{dx}{sin \hspace{.1cm} x \hspace{.1cm} cos \hspace{.1cm} x} = ln\left | tan \hspace{.1cm} x \right |

 

\int \frac{dx}{sin^{2} \hspace{.1cm} x \hspace{.1cm} cos \hspace{.1cm} x} = -\frac{1}{sin \hspace{.1cm} x}+ln\left | tan\left ( \frac{x}{2}+\frac{\Pi }{4} \right ) \right |

 

\int \frac{dx}{sin \hspace{.1cm} x \hspace{.1cm} cos^{2} \hspace{.1cm} x} = \frac{1}{cos \hspace{.1cm} x}+ln\left | tan\frac{x}{2} \right |

 

\int \frac{dx}{sin^{2} \hspace{.1cm} x \hspace{.1cm} cos^{2} \hspace{.1cm} x} = tan \hspace{.1cm} x-cot \hspace{.1cm} x

 

\int sin \hspace{.1cm} mx \hspace{.1cm} sin \hspace{.1cm} nxdx = -\frac{sin\left ( m+n \right )x}{2\left ( m+n \right )}+\frac{sin\left ( m-n \right )x}{2\left ( m-n \right )} \hspace{.1cm} \hspace{.1cm} \hspace{.1cm} m^{2}\neq n^{2}

 

\int sin \hspace{.1cm} mx \hspace{.1cm} cos \hspace{.1cm} nxdx = -\frac{cos\left ( m+n \right )x}{2\left ( m+n \right )}-\frac{cos\left ( m-n \right )x}{2\left ( m-n \right )} \hspace{.1cm} \hspace{.1cm} \hspace{.1cm} m^{2}\neq n^{2}

 

\int cos \hspace{.1cm} mx \hspace{.1cm} cos \hspace{.1cm} nxdx = \frac{sin\left ( m+n \right )x}{2\left ( m+n \right )}+\frac{sin\left ( m-n \right )x}{2\left ( m-n \right )} \hspace{.1cm} \hspace{.1cm} \hspace{.1cm} m^{2}\neq n^{2}

 

\int sin \hspace{.1cm} x \hspace{.1cm} cos^{n} \hspace{.1cm} xdx = -\frac{cos^{n+1}x}{n+1}

 

\int sin^{n} \hspace{.1cm} x \hspace{.1cm} cos \hspace{.1cm} xdx = \frac{sin^{n+1}x}{n+1}

 

\int arcsin \hspace{.1cm} xdx = x \hspace{.1cm} arcsin \hspace{.1cm} x + \sqrt{1-x^{2}}

 

\int arccos \hspace{.1cm} xdx = x \hspace{.1cm} arccos \hspace{.1cm} x - \sqrt{1-x^{2}}

 

\int arctan \hspace{.1cm} xdx = x \hspace{.1cm} arctan \hspace{.1cm} x-\frac{1}{2}ln\left ( x^{2}+1 \right )

 

\int arccot \hspace{.1cm} xdx = x \hspace{.1cm} arccot \hspace{.1cm} x+\frac{1}{2}ln\left ( x^{2}+1 \right )

 

 

 

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