Integrals Maths Formulas for Class 12 Chapter 7
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Formulas – Standard Integrals
- \(\int x^ndx=\frac{x^{n+1}}{n+1}+C,n\neq -1\). Particularly, \(\int dx=x+C)\)
- \(\int cos\:x\:dx=sin\:x+C\)
- \(\int sin\:x\:dx=-cos\:x+C\)
- \(\int sec^2x\:dx=tan\:x+C\)
- \(\int cosec^2x\:dx=-cot\:x+C\)
- \(\int sec\:x\:tan\:x\:dx=sec\:x+C\)
- \(\int cosec\:x\:cot\:x\:dx=-cosec\:x+C\)
- \(\int \frac{dx}{\sqrt{1-x^2}}=sin^{-1}x+C\)
- \(\int \frac{dx}{\sqrt{1-x^2}}=-cos^{-1}x+C\)
- \(\int \frac{dx}{1+x^2}=tan^{-1}x+C\)
- \(\int \frac{dx}{1+x^2}=-cot^{-1}x+C\)
- \(\int e^xdx=e^x+C\)
- \(\int a^xdx=\frac{a^x}{log\:a}+C\)
- \(\int \frac{dx}{x\sqrt{x^2-1}}=sec^{-1}x+C\)
- \(\int \frac{dx}{x\sqrt{x^2-1}}=-cosec^{-1}x+C\)
- \(\int \frac{1}{x}\:dx=log\:|x|+C\)
Formulas – Partial Fractions
Partial Fraction | Formulas |
\(\frac{px+q}{(x-a)(x-b)}\) | \(\frac{A}{x-a}+\frac{B}{x-b},a\neq b\) |
\(\frac{px+q}{(x-a)^2}\) | \(\frac{A}{x-a}+\frac{B}{(x-b)^2}\) |
\(\frac{px^2+qx+r}{(x-a)(x-b)(x-c)}\) | \(\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c}\) |
\(\frac{px^2+qx+r}{(x-a)^2(x-b)}\) | \(\frac{A}{x-a}+\frac{B}{(x-a)^2}+\frac{C}{x-b}\) |
\(\frac{px^2+qx+r}{(x-a)(x^2+bx+c)}\) | \(\frac{A}{x-a}+\frac{Bx+C}{x^2+bx+c}\) |
Formulas – Integration by Substitution
- \(\int tan\:x\:dx=log\:|sec\:x|+C\)
- \(\int cot\:x\:dx=log\:|sin\:x|+C\)
- \(\int sec\:x\:dx=log\:|sec\:x+tan\:x|+C\)
- \(\int cosec\:x\:dx=log\:|cosec\:x-cot\:x|+C\)
Formulas – Integrals (Special Functions)
- \(\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\:log\:\left |\frac{x-a}{x+a} \right |+C\)
- \(\int \frac{dx}{a^2-x^2}=\frac{1}{2a}\:log\:\left |\frac{a+x}{a-x} \right |+C\)
- \(\int \frac{dx}{x^2+a^2}=\frac{1}{a}\:tan^{-1}\frac{x}{a}+C\)
- \(\int \frac{dx}{\sqrt{x^2-a^2}}=log\:\left |x+\sqrt{x^2-a^2} \right |+C\)
- \(\int \frac{dx}{\sqrt{x^2+a^2}}=log\:\left |x+\sqrt{x^2+a^2} \right |+C\)
- \(\int \frac{dx}{\sqrt{x^2-a^2}}=sin^{-1}\frac{x}{a}+C\)
Formulas – Integration by Parts
- The integral of the product of two functions = first function × integral of the second function – integral of {differential coefficient of the first function × integral of the second function}
\(\int f_1(x).f_2(x)=f_1(x)\int f_2(x)\:dx-\int \left [ \frac{\mathrm{d} }{\mathrm{d} x}f_1(x).\int f_2(x)\:dx \right ]dx\) - \(\int e^x\left [ f(x)+f'(x) \right ]\:dx=\int e^x\:f(x)\:dx+C\)
Formulas – Special Integrals
- \(\int \sqrt{x^2-a^2}\:dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\:log\left | x+\sqrt{x^2-a^2} \right |+C\)
- \(\int \sqrt{x^2+a^2}\:dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\:log\left | x+\sqrt{x^2+a^2} \right |+C\)
- \(\int \sqrt{a^2-x^2}\:dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a}{2}\:sin^{-1}\frac{x}{a}+C\)
- \(ax^2+bx+c=a\left [ x^2+\frac{b}{a}x+\frac{c}{a} \right ]=a\left [ \left ( x+\frac{b}{2a} \right )^2+\left ( \frac{c}{a}-\frac{b^2}{4a^2} \right ) \right ]\)
Summary of Integrals formulas
We have listed top important formulas for Integrals for class 12 Chapter 7 which helps support to solve questions related to chapter Integrals. I would like to say that after remembering the Integrals formulas you can start the questions and answers solution of the Integrals chapter. If you faced any problem to find the solution of Integrals questions, please let me know through commenting or mail.
Maths Formulas for Class 12 by Chapters
Here Check Maths formulas for class 12 by chapter wise.
- Chapter 1 Relations and Functions
- Chapter 2 Inverse Trigonometric Functions
- Chapter 3 Matrices
- Chapter 4 Determinants
- Chapter 5 Continuity and Differentiability
- Chapter 7 Integrals
- Chapter 8 Applications of Integrals
- Chapter 10 Vector Algebra
- Chapter 11 Three dimensional Geometry
- Chapter 13 Probability