Home > Math Shortcuts > Integration Mathematical Formula

Integration Mathematical Formula

Last Modified on August 10, 2022 by Admin

Integration Mathematical Formula :

Here is the basic concept of Integration Mathematical Formula. These formulas will help you to solve math problems in competitive exams.

1. Common Integrals

—-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———

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

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

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

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

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

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

2. Integrals of Rational Functions
——————————————————-

——-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
——————————————————-

$\int xe^{cx}dx = \frac{e^{cx}}{c^{2}}\left ( cx-1 \right )$

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

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

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

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

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

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

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

4. Integrals of Logarithmic Functions
——————————————————-

$\int ln \hspace{.1cm} cxdx = x \hspace{.1cm} ln \hspace{.1cm} cx-x$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

5. Integrals of Trig. Functions
————————————————

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<< Go back to Mathematical Formula

We provided some other questions on other topics also. Click on the Next/Previous link to get more Question Answer session.

Please visit this page to get updates on more Math Shortcut Tricks and its uses.You can also like our facebook page to get updates.

If you have any question regarding Integration Mathematical Formula, then please do comment on below section.You can also send us message on facebook.

3 comments

tejas kadam says: August 22, 2018 at 2:03 pm
your skt trik is very useful for entrance exam

Reply
Mudit says: December 6, 2019 at 1:28 pm
When( a2 +b2 )=a2+b2+2ab+b2 so (X2+y2)=x2+2xy+y2

Reply
- Win MaC says: February 7, 2020 at 4:16 pm
  When (a+b)^2 =a^2+2ab+b^2
  (a^2+b^2) is different
  
  Reply

Disclaimer:

All contents of this website is fully owned by Math-Shortcut-Tricks.com. Any means of republish its content is strictly prohibited. We at Math-Shortcut-Tricks.com are always try to put accurate data, but few pages of the site may contain some incorrect data in it. We do not assume any liability or responsibility for any errors or mistakes in those pages. Visitors are requested to check correctness of a page by their own.