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Integration is widely used in engineering and science to find the area of curves, volume of solids, the lateral surface area of solids and other astrophysical applications.

Let F(x) be a differentiable function of ‘x’ such that d/dx[F(x)] = f(x) then the function F (x) is called a primitive or anti derivative or indefinite integral or simply integral of f(x) with respect to ‘x’ and is written symbolically as $\int$ f ( x) dx = F (x) + C or $\frac{d}{dx}$ [F(x) + C]= f(x)

Where f(x) is called the integrand and ‘C’ and is called the constant of Integration, may be real or imaginary.

Function | Integral of the function |

$\int$ x^{n} dx |
$\frac{(x^{n+1})}{( n + 1 )}$ + C |

$\int \frac{1}{2}$ dx | ln|x| + C |

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

$\int a^x$ dx |
$\frac{a^{x}}{In(a)}$+ C |

$\int$ 1 dx | x + C |

$\int$ cos (x) dx |
sin ( x ) + C |

$\int$ Tan (x) dx | - log | cos ( x ) | + C |

$\int$ Cot (x) dx | log | sin ( x ) | + C |

$\int$ Sec (x) dx | log | sec ( x ) + tan ( x ) | + C |

$\int$ Csc (x) dx | log | css ( x ) - cot ( x ) | + C |

$\int$ Sin (x) dx | - cos ( x ) + C |

$\int\frac{1}{\sqrt{a^2-x_2}}$ dx | sin-1$\frac{1}{a}$ + C |

$\int\frac{-1}{\sqrt{a^2-x_2}}$ dx | cos-1($\frac{x}{a}$) + C |

$\int\frac{1}{a^2+x_2}$ dx | tan-1($\frac{x}{a}$) + C |

$\int\frac{-1}{a^2+x_2}$ dx | cot-1($\frac{x}{a}$) + C |

$\int\frac{-1}{|x|\sqrt{a^2-x_2}}$ dx |
(1/a) * csc-1($\frac{x}{a}$) + C |

$\int\frac{1}{|x|\sqrt{a^2-x_2}}$ dx | (1/a) * sec-1($\frac{x}{a}$) + C |

$\int$ sinh (x) dx | cosh ( x ) + |

$\int$ cosh ( x ) dx | sinh ( x ) + C |

$\int$ tanh ( x ) dx | ln cosh ( x ) + C |

$\int$ coth ( x ) dx | ln |sinh ( x )| + C |

$\int$ Sech ( x ) dx | tan-1 |sinh ( x )| + C |

$\int$ Csch^{2} ( x ) dx |
-coth ( x ) + C |

- Integration of 1 is equal to ‘ x ’ + C

- Integration of xn is equal to ‘ $\frac{(xn + 1)}{(n + 1)}$ + C ‘ where n is not equal to 1

- Integration of $\frac{1}{x}$ is equal to ‘ ln |x| ’ + C

- Integration of ($\frac{1}{ax + b}$) is equal to ‘ $\frac{1}{a}$ * ln (ax + b) ’ + C

- Integration of
**sin ( x )**is equal to ‘ -**cos ( x )**’ + C - Integration of cos ( x ) is equal to ‘ sin ( x ) ’ + C
- Integration of tan ( x ) is equal to ‘- log | cos ( x ) | ’ + C
- Integration of cot ( x ) is equal to ‘ log | sin ( x ) | ’ + C
- Integration of sec ( x ) is equal to ‘ log | sec ( x ) + tan ( x ) | ’ + C
- Integration of csc ( x ) is equal to ‘ log | css ( x ) - cot ( x ) |’ + C
- Integration of sec
^{2}( x ) is equal to ‘ tan ( x ) ’ + C - Integration of csc
^{2}( x ) is equal to ‘ - cot ( x ) ’ + C - Integration of sec ( x ) tan( x ) is equal to ‘ sec ( x ) ’ + C
- Integration of csc ( x ) cot ( x ) is equal to ‘ - csc ( x ) ’ + C

D[u(x) * v(x) ] = u(x) * v’(x) + v(x) * u’(x)

Integrating both sides with respect to ‘x’, we get

uv = $\int u \frac{dv}{dx}$dx+$\int v \frac{du}{dx}$ dx

= $\int u \frac{dv}{dx}$ dx = uv - $\int u \frac{du}{dx}$

- - - - - - - - - - - - - - - - - - - - - - - (i)

Now put u = f( x ) and v = g ( x ) dx, so that $\frac{dv}{dx}$ = g( x )

Substituting these values in equation (i), we get

f( x). g( x ) dx = f( x ) * {$\int$ g (x) dx} - $\int$ [f’ ( x ) * {$\int$ g ( x ) dx}] dx

The integral of the product of two functions = (first function) x (integral of the second function) – Integral of the derivative of the first function x Integral of the second function)

If y(x) is one of the primitives or anti derivatives of a function f (x) defined on [a, b], then the definite integral of f(x) over [a, b] is given by y (b) – y (a) and is denoted by $\int_{a}^{b} f(x) dx$ read as integral of f(x) dx from ‘a’ to ‘b’.

The numbers ‘a’ and ‘b’ are called the limits of integration ‘a’ is called the lower limit and ‘b’ is called the upper limit. The interval [a, b] is called the interval of integration.

If f ( x ) is continuous on [a,∞ ] then $\int_{0}^{\infty } f(x) dx$ is called an improper integral and is defined as

$\int_{0}^{\infty } f(x) dx$ = $\lim_{b \to \infty } \int_{a}^{b}f(x) dx$ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (i)

This above equation is read as integral f(x) dx from ‘0’ to infinity is equal to limit b tends to infinity integral f(x) dx from a to b

If there exists a finite limit on the right hand side of (i) then we say that the improper integral is convergent, otherwise it is divergent

Similarly, we define $\int_{0}^{-\infty } f(x) dx$ = $\lim_{b \to \infty } \int_{a}^{b}f(x) dx$ and $\int_{\infty}^{-\infty } f(x) dx$ = $\int_{\infty}^{-\infty } f(x) dx+\int_{a}^{-\infty } f(x) dx$

Geometrically, for f ( x ) > 0 the improper $\int_{a}^{-\infty } f(x) dx$ integral gives area of the figure bounded by the curve y = f ( x ), the x – axis and the straight line x = a.

$\int$ e

(ii) Following are the trigonometric substitutions that are useful in evaluating integrals

S. No |
Expression |
Substitution |

1 |
a^{2} + x^{2} |
x = a tanθ or x = a cotθ |

2 |
a^{2} - x^{2} |
x = a sinθ or x = a cosθ |

3 |
x^{2} - a^{2} |
x = a secθ or x = a cscθ |

4 |
$\sqrt{[\frac{(a – x)}{(a + x)}]}$or$\sqrt{[\frac{(a + x)}{(a - x)}]}$ |
x = a cos^{2}θ |

5 |
$\sqrt{[\frac{(x-p)}{(q- x)}]}$ |
x = p cos^{2}θ + q sin^{2}θ |

Here are some special integrals formula:

- Integration of 1 divided by square root of ( a
^{2}– x^{2}) is equal to ‘ sin-1($\frac{x}{a}$) ’ + C

- Integration of - 1 divided by square root of ( a
^{2}– x^{2}) is equal to ‘ cos-1($\frac{x}{a}$) ’ + C

- Integration of 1 divided by ( a
^{2}+ x^{2}) is equal to ‘ 1/a * tan-1($\frac{x}{a}$) ’ + C

- Integration of - 1 divided by ( a
^{2}+ x^{2}) is equal to ‘ cot-1($\frac{x}{a}$) ’ + C

- Integration of 1 divided by x times square root of ( x
^{2}– a^{2}) is equal to ‘ ($\frac{1}{a}$) *sec-1($\frac{x}{a}$) ’ + C

- Integration of - 1 divided by x times square root of ( x
^{2}– a^{2}) is equal to ‘ (1/a) *csc-1($\frac{x}{a}$) ’ + C

- Integration of 1 divided by ( a
^{2}- x^{2}) is equal to ‘ $\frac{1}{2}$a * |$\frac{log|(x - a)}{(x + a)}$|’ + C

- Integration of 1 divided by ( x
^{2}- a^{2}) is equal to ‘ $\frac{1}{2a}$ * |$\frac{log|(a+x)}{(a-x)}$| ’ + C

- Integration of 1 divided by $\sqrt{a^2+x^2}$ is equal to ‘ log | x + $\sqrt{ (a^2 + x^2)}$ | ’ + C

- Integration of 1 divided by $\sqrt{x^2-a^2}$ is equal to ‘ log | x + $\sqrt (x^2 - a^2)$ | ’ + C

- Integration of $\sqrt{a^2-x^2}$ is equal to ‘ $\frac{1}{2}x$ $\sqrt(a^2 - x^2)+\frac{1}{2}a2 sin-1\frac{x}{a}$ ’ + C

- Integration of $\sqrt({a^2+x^2)}$ is equal to ‘ $\frac{1}{2}x\sqrt(a^2+x^2 )+\frac{1}{2}a2 log | x + \sqrt(a^2+ x^2)$ |’ + C

- Integration of $\sqrt{x^2-a^2}$ is equal to ‘ $\frac{1}{2}x \sqrt(x^2-a^2 ) - \frac{1}{2}a2 log | x + \sqrt(x^2-a^2)$ |’ + C

- Integration of sinh ( x ) is equal to ‘ cosh ( x ) ’ + C

- Integration of cosh (x) is equal to ‘ sinh ( x ) ’ + C

- Integration of tanh ( x ) is equal to ‘ ln cosh ( x ) ’ + C

- Integration of coth ( x ) is equal to ‘ ln |sinh ( x )| ’ + C

- Integration of sech ( x ) is equal to ‘ tan-1 |sinh ( x )| ’ + C

- Integration of sech( x ).tanh( x ) is equal to ‘ - sech ( x ) ’ + C

- Integration of cssh( x ).coth( x ) is equal to ‘ - csch ( x ) ’ + C

1. $\int_{a}^{b}(f(x)+g(x))+dx$ =$\int_{a}^{b} f(x) dx$ = $\int_{a}^{b} g(x) dx$

2. $\int_{a}^{b} k. f(x) dx$ = $k\int_{a}^{b} f(x) dx$

3. $\int_{a}^{b} f(x) dx$ = $\int_{a}^{b} f(t) dt$

4. $\int_{a}^{b} f(x) dx$ = $\int_{a}^{b} f(x) dx$

5. $\int_{a}^{b} f(x) dx$ = $\int_{a}^{c} f(x) dx$ + $\int_{c}^{b} f(x) dx$

6. $\int_{a}^{b} f(x) dx$ = $\int_{a}^{b} f(a+b-x) dx$

7. $\int_{0}^{a} f(x) dx$ = $\int_{0}^{a} f(a-x) dx$

8. $\int_{a}^{b} f(x) dx$ = $\int_{a-c}^{b-c} f(x+c) dx$

$\int_{a}^{b} f(x) dx$ = $\int_{a+c}^{b+c} f(x-c) dx$

9. $\int_{a}^{b} f(x) dx$ = $k\int_{\frac{a}{k}}^{\frac{b}k{}} f(kx)dx$

$\int_{a}^{b} f(x) dx$ = $\frac{1}{k} \int_{a}^{b} f(\frac{x}{k}) dx$

10. $\int_{a}^{b} f(x) dx$ = $\int_{-a}^{-b} f(-x) dx$

11. $\int_{0}^{2a} f(x) dx$ = $\int_{0}^{a} {f(a-x)+f(a+x)} dx$

12. $\int_{a}^{b} f(x) dx$ = $(b-a)\int_{0}^{1} {f(b-a)x+a} dx$

2. Triple Integrals

3. Line Integrals

4. Surface Integrals

The concept of the definite integral could be extended to functions that have more than one variable.

Let us take, a function of two variables z = f (x, y). The double integral of this function f (x, y) is written as

$\int\int_{p} f(x,y) da $

Where ‘ P ‘ is the region of integration in the xy - plane.

If $\int_{a}^{b}f(x) dx$ is the definite integral of a function f(x) in one variable such that f (x) ≥ 0 is the area under the curve f (x) from x = a to x = b, then the double integral will give the volume under the surface of z = f (x, y) and the xy - plane in the region of integration ‘P’.

Triple integrals

Double integration is used to integrate over a two dimensional region, therefore a triple integral is used to integrate over a three dimensional region.

$\int\int_{A}\int f(x,y,z) dv$

Where, A = [a, b] x [c, d] x [e, f]

That is = $\int \int_{A} \int f(x,y,z)dv$ = $\int_{e}^{f}\int_{c}^{d}\int_{a}^{b}f(x,y,z)dx dy dz$

Here we first integrate the function with respect to ‘x’ then with respect to ‘y’ and then with respect to ‘z’

(i) Area under a curve

(ii) Area between two curves

(iii) Volumes of solids with known cross sections

(iv) Volumes of solids of revolution ( using disks and washers)

(v) Arc length

(vi) Area and volumes involving parametrically defined functions

(vii) Area and arc length for polar curves

If ‘f’ be a continuous function of ‘x’ defined in the closed interval [a, b] and the function F is an anti derivative of ‘f’ on [a, b], such that F’ ( x ) = f( x ) for all ‘x’ in the domain of ‘f’, then $\int_{a}^{b} f(x) dx$ = [F ( x ) + C ]ba

= F ( b ) + C – [F ( a ) + C]

= F( b ) + C – F( a ) – C

= F( b ) – F( a )

$\frac{d}{dx} \int_{x}^{a} f(t) dt $=f(x)

In Numerical Integration first we approximate the integrand ‘f’ by a polynomial ‘p’ and then find $\int_{a}^{b} p(x) dx$

Then $\int_{a}^{b} f(x) dx$ is approximately equal to $\int_{a}^{b} p(x) dx$. The absolute difference | $\int_{a}^{b} f(x) dx-\int_{a}^{b} p(x) dx$| is called the error term in such approximation. There are several methods for approximating a definite integral, such as

Then, $\int_{a}^{b} f(x) dx$ =$(\frac{h}{2})$(y0 + y1) + $(\frac{h}{2})$(y1 + y2) + $(\frac{h}{2})$(y2 + y3) + . . . . . . . . . . . . . . . + $(\frac{h}{2})$(yn - 1 + yn)

$\int_{a}^{b} f(x) dx$ = $(\frac{h}{2})$* [(y0 + yn) + 2(y1 + y2 + y3 + . . . . . . . . . . . . .. . . + yn - 2)]

Step 1: Note down a, b, and n the number of divisions

Step 2: Calculate the value h =$\frac{(b-a)}{n}$

Step 3: Compute the values of ‘f’ at x = x

Step 4: Write down the Trapezoidal rule for the division specified in the question and substitute the values obtained in step 3 in the rule 91) and simplify.

Definite IntegralsImproper IntegralsIntegration Rules Numerical IntegrationTable of IntegralsCommon Integrals Indefinite IntegralsTriple IntegralsFundamental Theorem of Calculus Integration FormulasIntegration by PartsTrigonometric Integrals Trapezoidal RuleTrigonometric SubstitutionDouble Integrals